From 0f2e69d1f9affd0f3efab2a5fa81549d8c77b69f Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Tue, 5 May 2026 01:50:35 -0400 Subject: [PATCH] Polished A-A and added new lines for broken enumerates. --- src/cat/cat/cat-func.tex | 3 +- src/cat/cat/tensor.tex | 6 ++- src/cat/cat/universal.tex | 12 ++++-- src/cat/gluing/index.tex | 9 ++-- src/cat/tricks/dyadic.tex | 3 +- src/dg/derivative/higher.tex | 3 +- src/dg/derivative/mvt.tex | 6 ++- src/dg/derivative/sets.tex | 12 ++++-- src/fa/duality/polar.tex | 3 +- src/fa/lc/bornologic.tex | 3 +- src/fa/lc/convex.tex | 9 ++-- src/fa/lc/hahn-banach.tex | 3 +- src/fa/lc/inductive.tex | 15 ++++--- src/fa/lc/quotient.tex | 3 +- src/fa/lc/spaces-of-linear.tex | 3 +- src/fa/lc/tensor.tex | 12 ++++-- src/fa/lp/definition.tex | 3 +- src/fa/norm/absolute.tex | 6 ++- src/fa/norm/multilinear.tex | 3 +- src/fa/norm/normed.tex | 9 ++-- src/fa/norm/separable.tex | 3 +- src/fa/order/lattice.tex | 18 +++++--- src/fa/rs/bv.tex | 6 ++- src/fa/rs/rs-bv.tex | 9 ++-- src/fa/tvs/complete-metric.tex | 24 +++++++---- src/fa/tvs/completion.tex | 6 ++- src/fa/tvs/continuous.tex | 6 ++- src/fa/tvs/definition.tex | 15 ++++--- src/fa/tvs/dual.tex | 3 +- src/fa/tvs/inductive.tex | 6 ++- src/fa/tvs/metric.tex | 6 ++- src/fa/tvs/projective.tex | 9 ++-- src/fa/tvs/quotient.tex | 6 ++- src/fa/tvs/spaces-of-linear.tex | 6 ++- src/measure/bochner-integral/bochner.tex | 9 ++-- src/measure/bochner-integral/strongly.tex | 6 ++- src/measure/lebesgue-integral/complex.tex | 3 +- src/measure/measurable-maps/metric.tex | 9 ++-- src/measure/measurable-maps/real-valued.tex | 3 +- src/measure/measure/complete.tex | 3 +- src/measure/measure/kolmogorov.tex | 12 ++++-- src/measure/measure/lebesgue-stieltjes.tex | 3 +- src/measure/measure/measure.tex | 15 ++++--- src/measure/measure/outer.tex | 3 +- src/measure/measure/product.tex | 12 ++++-- src/measure/measure/regular.tex | 3 +- src/measure/measure/semifinite.tex | 3 +- src/measure/measure/sigma-finite.tex | 3 +- src/measure/radon/c0.tex | 3 +- src/measure/radon/radon.tex | 3 +- src/measure/radon/riesz.tex | 3 +- src/measure/sets/algebra.tex | 3 +- src/measure/sets/borel.tex | 3 +- src/measure/sets/elementary.tex | 3 +- src/measure/sets/lambda.tex | 3 +- src/measure/vector/complex.tex | 9 ++-- src/measure/vector/variation.tex | 3 +- src/measure/vector/vector.tex | 3 +- src/process/markov/definition.tex | 3 +- src/topology/functions/set-systems.tex | 9 ++-- src/topology/functions/uniform.tex | 12 +++++- src/topology/main/baire.tex | 6 ++- src/topology/main/compact.tex | 6 ++- src/topology/main/connected.tex | 3 +- src/topology/main/continuity.tex | 9 ++-- src/topology/main/definition.tex | 47 +++++++++++++++++++-- src/topology/main/filters.tex | 15 ++++--- src/topology/main/hausdorff.tex | 3 +- src/topology/main/lch.tex | 46 +++++++++++++++----- src/topology/main/neighbourhoods.tex | 6 ++- src/topology/main/normal.tex | 3 +- src/topology/main/product.tex | 2 +- src/topology/main/quotient.tex | 6 ++- src/topology/main/regular.tex | 3 +- src/topology/metric/metric.tex | 3 +- src/topology/uniform/cauchy.tex | 3 +- src/topology/uniform/completion.tex | 15 ++++--- src/topology/uniform/definition.tex | 15 ++++--- src/topology/uniform/equicontinuous.tex | 15 +++++++ src/topology/uniform/metric.tex | 18 +++++--- src/topology/uniform/uc.tex | 6 ++- 81 files changed, 441 insertions(+), 185 deletions(-) diff --git a/src/cat/cat/cat-func.tex b/src/cat/cat/cat-func.tex index 5de70eb..5397796 100644 --- a/src/cat/cat/cat-func.tex +++ b/src/cat/cat/cat-func.tex @@ -13,7 +13,8 @@ \item[(CAT1)] For any $A, B, A', B' \in \obj{\catc}$, $\mor{A, B}$ and $\mor{A', B'}$ are disjoint or equal, where $\mor{A, B} = \mor{A', B'}$ if and only if $A = A'$ and $B = B'$. \item[(CAT2)] For any $A \in \obj{\catc}$, there exists $\text{Id}_A \in \mor{A, A}$ such that $f \circ \text{Id}_A = f$ and $\text{Id}_A \circ g = g$ for all $B, C \in \obj{\catc}$, $f \in \mor{A, B}$, and $g \in \mor{C, A}$. \item[(CAT3)] For any $A, B, C, D \in \obj{\catc}$, $f \in \mor{A, B}$, $g \in \mor{B, C}$, and $h \in \mor{C, D}$, $(h \circ g) \circ f = h \circ (g \circ f)$. - \end{enumerate} + \end\{enumerate\} + The elements of $\obj{\catc}$ are the \textbf{objects} of $\catc$, and elements of $\mor{A, B}$ are the \textbf{morphisms/arrows} from $A$ to $B$. \end{definition} diff --git a/src/cat/cat/tensor.tex b/src/cat/cat/tensor.tex index 7d8757e..6fbb967 100644 --- a/src/cat/cat/tensor.tex +++ b/src/cat/cat/tensor.tex @@ -32,7 +32,8 @@ A_i \ar@{->}[u]^{\iota_i} \ar@{->}[ru]_{T_i} & } \] - \end{enumerate} + \end\{enumerate\} + The module $A = \bigoplus_{i \in I}A_i$ is the \textbf{direct sum} of $\seqi{A}$. \end{definition} @@ -204,7 +205,8 @@ \[ (x_1, \cdots, \alpha x_j, \cdots, x_n) - \alpha(x_1, \cdots, x_n) \] - \end{enumerate} + \end\{enumerate\} + (1), (2): Let $\bigotimes_{j = 1}^n E_j = M/N$ and \[ diff --git a/src/cat/cat/universal.tex b/src/cat/cat/universal.tex index 784ba85..673bb28 100644 --- a/src/cat/cat/universal.tex +++ b/src/cat/cat/universal.tex @@ -7,7 +7,8 @@ \begin{enumerate} \item \textbf{universally attracting} if for every $A \in \obj{\catc}$, there exists a unique $f \in \mor{A, P}$. \item \textbf{universally repelling} if for every $A \in \obj{\catc}$, there exists a unique $f \in \mor{P, A}$. - \end{enumerate} + \end\{enumerate\} + If $P$ is universally attracting or repelling, then $P$ is a \textbf{universal object}. If $P, Q \in \obj{\catc}$ are both universally attracting/repelling, then they are isomorphic. @@ -62,12 +63,14 @@ \begin{enumerate} \item For any $i \in I$, $i \lesssim i$. \item For any $i, j, k \in I$ such that $i \lesssim j$ and $j \lesssim k$, $i \lesssim k$. - \end{enumerate} + \end\{enumerate\} + and one of the following holds: \begin{enumerate} \item[(3U)] For any $i, j \in I$, there exists $k \in I$ with $i, j \lesssim k$. \item[(3D)] For any $i, j \in I$, there exists $k \in I$ with $k \lesssim i, j$. - \end{enumerate} + \end\{enumerate\} + The directed set is \textbf{upward-directed} if it satisfies (3U), and \textbf{downward-directed} if it satisfies (3D). \end{definition} @@ -85,7 +88,8 @@ \item For each $i \in I$, $f^i_i = \text{Id}_{A_i}$. \item For each $i, j \in I$ with $i \lesssim j$, $f^i_j \in \mor{A_i, A_j}$. \item For each $i, j, k \in I$ with $i \lesssim j \lesssim k$, $f^j_k \circ f^i_j = f^i_k$. - \end{enumerate} + \end\{enumerate\} + If $I$ is upward/downward-directed, then $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ is upward/downward-directed. \end{definition} diff --git a/src/cat/gluing/index.tex b/src/cat/gluing/index.tex index c235213..76396c0 100644 --- a/src/cat/gluing/index.tex +++ b/src/cat/gluing/index.tex @@ -8,7 +8,8 @@ \begin{enumerate} \item[(a)] $\bigcup_{i \in I}U_i = X$. \item[(b)] For each $i, j \in I$, either $U_i \cap U_j = \emptyset$, or $f_i|_{U_i \cap U_j} = f_j|_{U_i \cap U_j}$. - \end{enumerate} + \end\{enumerate\} + then there exists a unique $f: X \to Y$ such that $f|_{U_i} = f_i$ for all $i \in I$. \end{lemma} \begin{proof} @@ -16,7 +17,8 @@ \begin{enumerate} \item By assumption (a), $\bracs{x|(x, y) \in \Gamma} = \bigcup_{i \in I}U_i = X$. \item For any $x \in X$, there exists $y \in Y$ with $(x, y) \in \Gamma$, and $i \in I$ such that $(x, y) \in \Gamma_i$. If $(x, y') \in \Gamma_j \subset \Gamma$, then $x \in U_i \cap U_j \ne \emptyset$. By assumption (b), $y = y'$. - \end{enumerate} + \end\{enumerate\} + Thus $\Gamma$ is the graph of a function $f: X \to Y$ with $f|_{U_i} = f_i$ for all $i \in I$. \end{proof} @@ -27,7 +29,8 @@ \item[(a)] $\bigcup_{V \in \fF}V = E$. \item[(b)] For each $V, W \in \fF$, $T_V|_{V \cap W} = T_W|_{V \cap W}$. \item[(c)] $\fF$ is upward-directed with respect to includion. - \end{enumerate} + \end\{enumerate\} + then there exists a unique $T \in \hom(E; F)$ such that $T|_{V} = T_V$ for all $V \in \fF$. \end{lemma} \begin{proof} diff --git a/src/cat/tricks/dyadic.tex b/src/cat/tricks/dyadic.tex index 040d6f5..3ce7166 100644 --- a/src/cat/tricks/dyadic.tex +++ b/src/cat/tricks/dyadic.tex @@ -42,7 +42,8 @@ \begin{enumerate} \item[(a)] for each $n \in \natp$, $g_{n+1} + g_{n+1} \le g_n$. \item[(b)] For each $x, y \in G$, $x + y \ge x, y$. - \end{enumerate} + \end\{enumerate\} + For each $x \in \mathbb{D} \cap [0, 1)$, let $\phi(x) = \sum_{n \in M(x)}g_n$, then \begin{enumerate} diff --git a/src/dg/derivative/higher.tex b/src/dg/derivative/higher.tex index deeeb64..35c32cb 100644 --- a/src/dg/derivative/higher.tex +++ b/src/dg/derivative/higher.tex @@ -9,7 +9,8 @@ \begin{enumerate} \item There exists $V \in \cn_E(x_0)$ such that $f$ is $(n-1)$-fold differentiable on $V$. \item The derivative $D_\sigma^{n-1}f: U \to B^{n-1}_\sigma(E; F)$ is derivative at $x_0$. - \end{enumerate} + \end\{enumerate\} + In which case, $D_\sigma(D_\sigma^{n-1}f)(x_0) \in L(E; B^{n-1}_\sigma(E; F))$ is the \textbf{$n$-fold $\sigma$-derivative of $f$ at $x_0$}. The mapping $f: U \to F$ is \textbf{$n$-fold $\sigma$-differentiable on $U$} if it is $n$-fold $\sigma$-differentiable at every point in $U$. Under the identification $B_\sigma(E; B^{n-1}_\sigma(E; F)) = B_\sigma^{n}(E; F)$ given by \autoref{proposition:multilinear-identify}, diff --git a/src/dg/derivative/mvt.tex b/src/dg/derivative/mvt.tex index 2ee3e24..1c2b236 100644 --- a/src/dg/derivative/mvt.tex +++ b/src/dg/derivative/mvt.tex @@ -17,7 +17,8 @@ \begin{enumerate} \item[(a)] $f, g$ are right-differentiable on $[a, b] \setminus N$. \item[(b)] For every $x \in [a, b] \setminus N$, $D^+f(x) \le D^+g(x)$. - \end{enumerate} + \end\{enumerate\} + then for any $x \in [a, b]$, $f(x) - f(a) \le g(x) - g(a)$. \end{lemma} \begin{proof} @@ -51,7 +52,8 @@ \item $f, g$ are right-differentiable on $[a, b] \setminus N$. \item For each $x \in [a, b] \setminus N$, $D^+f(x) \in D^+g(x)B$. \item $g$ is non-decreasing. - \end{enumerate} + \end\{enumerate\} + then \[ f(b) - f(a) \in [g(b) - g(a)]B diff --git a/src/dg/derivative/sets.tex b/src/dg/derivative/sets.tex index ad647a4..0c1e03e 100644 --- a/src/dg/derivative/sets.tex +++ b/src/dg/derivative/sets.tex @@ -8,7 +8,8 @@ \item For each $A \in \sigma$, $r(th)/t^n \to 0$ uniformly on $A$. \item If $r_t(x) = r(tx)/t^n$, then $r_t \to 0$ as $t \to 0$ with respect to the $\sigma$-uniform topology on $F^E$. \item For each $A \in \sigma$, $\seq{a_k} \subset A$, and $\seq{t_k} \subset K \setminus \bracs{0}$ with $t_k \to 0$ as $n \to \infty$, $r(t_ka_k)/t_k^n \to 0$ as $n \to \infty$. - \end{enumerate} + \end\{enumerate\} + If the above holds, then $r$ is \textbf{$\sigma$-small of order $n$}. The set $\mathcal{R}_\sigma^n(E; F)$ is the $K$-vector space of all $\sigma$-small functions of order $n$ from $E$ to $F$. For simplicity, $\mathcal{R}_\sigma(E; F)$ denotes $\mathcal{R}_\sigma^1(E; F)$. @@ -52,7 +53,8 @@ \begin{enumerate} \item[(a)] For any $r \in \mathcal{R}_\sigma(E; F)$ and $T \in L(F; G)$, $T \circ r \in \mathcal{R}_\sigma(E; G)$. \item[(b)] For any $r \in \mathcal{R}_\sigma(E; F)$, $T \in L(E; F)$, and $s \in \mathcal{R}_\tau(F; G)$, $s \circ (T + r) \in \mathcal{R}_\sigma(E; G)$. - \end{enumerate} + \end\{enumerate\} + then for any $U \subset E$ and $V \subset F$ open, $f: U \to V$ $\sigma$-differentiable at $x_0 \in U$, $g: V \to F$ $\tau$-differentiable at $f(x_0) \in V$, $g \circ f: U \to F$ is $\sigma$-differentiable at $x_0$ with \[ D_\sigma(g \circ f)(x_0) = D_\tau g(f(x_0)) \circ D_\sigma f(x_0) @@ -84,12 +86,14 @@ \begin{enumerate} \item Compact sets. \item Bounded sets. - \end{enumerate} + \end\{enumerate\} + then \begin{enumerate} \item For any $r \in \mathcal{R}_\sigma(E; F)$ and $T \in L(F; G)$, $T \circ r \in \mathcal{R}_\sigma(E; G)$. \item For any $r \in \mathcal{R}_\sigma(E; F)$, $T \in L(E; F)$, and $s \in \mathcal{R}_\tau(F; G)$, $s \circ (T + r) \in \mathcal{R}_\sigma(E; G)$. - \end{enumerate} + \end\{enumerate\} + and by \autoref{proposition:chain-rule-sets}, $\sigma$-derivatives and $\tau$-derivatives satisfy the Chain rule. \end{proposition} \begin{proof} diff --git a/src/fa/duality/polar.tex b/src/fa/duality/polar.tex index fbfe1c9..5b4ecef 100644 --- a/src/fa/duality/polar.tex +++ b/src/fa/duality/polar.tex @@ -27,7 +27,8 @@ \begin{enumerate} \item $\emptyset^\circ = \emptyset^\square = F$ and $F^\circ = F^\square = \bracs{0}$. \item For any $A, B \subset E$ and $\lambda \ne 0$, if $\lambda A \subset B$, then $B^\circ \subset \lambda^{-1}A^\circ$. - \end{enumerate} + \end\{enumerate\} + \end{proposition} diff --git a/src/fa/lc/bornologic.tex b/src/fa/lc/bornologic.tex index 0341051..1792823 100644 --- a/src/fa/lc/bornologic.tex +++ b/src/fa/lc/bornologic.tex @@ -7,7 +7,8 @@ \begin{enumerate} \item For any $U \subset E$ convex and balanced, if $U$ absorbs every bounded set of $E$, then $U \in \cn_E(0)$. \item For any seminorm $\rho: E \to [0, \infty)$ that is bounded on all bounded sets of $E$, $\rho$ is continuous. - \end{enumerate} + \end\{enumerate\} + If the above holds, then $E$ is a \textbf{bornological space}. \end{definition} diff --git a/src/fa/lc/convex.tex b/src/fa/lc/convex.tex index 08b30ca..88c9d79 100644 --- a/src/fa/lc/convex.tex +++ b/src/fa/lc/convex.tex @@ -138,7 +138,8 @@ \item For each $i \in I$, $d_i: E \times E \to [0, \infty)$ defined by $(x, y) \mapsto [x - y]_i$ is a pseudo-metric. \item The topology induced by $\seqi{d}$ makes $E$ a topological vector space. \item For each $i \in I$, $[\cdot]_i: E \to [0, \infty)$ is continuous. - \end{enumerate} + \end\{enumerate\} + The topology induced by $\seqi{d}$ is the \textbf{vector space topology induced by} $\seqi{[\cdot]}$. In addition, \begin{enumerate} \item[(U)] For any family $\bracsn{[\cdot]_j}_{j \in J}$ of continuous seminorms on $E$, the vector space topology induced by $\bracsn{[\cdot]_j}_{j \in J}$ is contained in the vector space topology induced by $\seqi{[\cdot]}$. @@ -159,7 +160,8 @@ \item If $A$ is convex, then for any $x, y \in E$, $[x + y]_A \le [x]_A + [y]_A$. \item If $A$ is circled, then for any $x \in E$ and $\lambda \in K$, $[\lambda x]_A = \abs{\lambda}[x]_A$. \item If $A$ is circled, then $\bracs{\rho < 1} \subseteq A \subseteq \bracs{\rho \le 1} \subseteq \ol A$. - \end{enumerate} + \end\{enumerate\} + In particular, \begin{enumerate}[start=4] \item If $A$ is convex, then $[\cdot]_A$ is a sublinear functional. @@ -189,7 +191,8 @@ \item There exists a fundamental system of neighborhoods at $0$ consisting of convex sets. \item There exists a fundamental system of neighbourhoods at $0$ consisting of convex, circled, and radial sets. \item There exists a family of seminorms $\seqi{[\cdot]}$ that induces the topology on $E$. - \end{enumerate} + \end\{enumerate\} + If the above holds, then $E$ is a \textbf{locally convex} space. \end{definition} \begin{proof} diff --git a/src/fa/lc/hahn-banach.tex b/src/fa/lc/hahn-banach.tex index fa998ad..94b7172 100644 --- a/src/fa/lc/hahn-banach.tex +++ b/src/fa/lc/hahn-banach.tex @@ -134,7 +134,8 @@ \begin{enumerate} \item $|\phi| \le \rho$. \item $\dpb{x, \phi}{E} = \inf_{y \in M}\rho(x + y)$. - \end{enumerate} + \end\{enumerate\} + \item For any $x \in E$ and continuous seminorm $\rho: E \to [0, \infty)$, there exists $\phi \in E^*$ with $|\phi| \le \rho$ and $\dpb{x, \phi}{E} = \rho(x)$. \item If $E$ is separated, then for any $x, y \in E$, there exists $\phi \in E^*$ with $\dpb{x, \phi}{E} \ne \dpb{y, \phi}{E}$. \end{enumerate} diff --git a/src/fa/lc/inductive.tex b/src/fa/lc/inductive.tex index cee34ed..50ab64b 100644 --- a/src/fa/lc/inductive.tex +++ b/src/fa/lc/inductive.tex @@ -21,7 +21,8 @@ \] is a fundamental system of neighbourhoods for $E$ at $0$. - \end{enumerate} + \end\{enumerate\} + The topology $\topo$ is the \textbf{inductive locally convex topology} on $E$ induced by $\seqi{T}$. \end{definition} \begin{proof} @@ -64,7 +65,8 @@ \] is a fundamental system of neighbourhoods for $E$ at $0$. - \end{enumerate} + \end\{enumerate\} + The space $E = \bigoplus_{i \in I}E_i$ is the \textbf{locally convex direct sum} of $\seqi{E}$. \end{definition} @@ -110,7 +112,8 @@ \] is a fundamental system of neighbourhoods for $E$ at $0$. - \end{enumerate} + \end\{enumerate\} + The pair $(E, \bracsn{T^i_E}_{i \in I})$ is the \textbf{inductive limit} of $(\seqi{E}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$. \end{definition} \begin{proof} @@ -174,7 +177,8 @@ \begin{enumerate} \item[(a)] $B$ is bounded. \item[(b)] There exists $n \in \natp$ such that $B \subset E_n$ is bounded. - \end{enumerate} + \end\{enumerate\} + \item If $E_n$ is complete for each $n \in \natp$, then $E$ is also complete. \end{enumerate} \end{proposition} @@ -190,7 +194,8 @@ \item For each $k \in \natp$, $U_k \in \cn_{E_{n_k}}(0)$. \item For each $k \in \natp$, $U_k = U_{k+1} \cap E_{n_k}$. \item For each $k \in \natp$, $n^{-1}x_k \not\in U_k$. - \end{enumerate} + \end\{enumerate\} + then $V = \bigcup_{k \in \natp}U_k \in \cn_E(0)$ with $V \cap E_{n_k} = U_k$ for all $k \in \natp$. For any $n \in \natp$, $x_k \not\in nU_k = nV \cap E_{n_k}$. Therefore $B$ is not bounded. (3), $(b) \Rightarrow (a)$: Let $U \in \cn_E(0)$, then $U \cap E_n \in \cn_{E_n}(0)$, so there exists $\lambda \in K$ with $\lambda (U \cap E_n) \supset B$. diff --git a/src/fa/lc/quotient.tex b/src/fa/lc/quotient.tex index 0f3113a..14ebe11 100644 --- a/src/fa/lc/quotient.tex +++ b/src/fa/lc/quotient.tex @@ -41,7 +41,8 @@ If $F$ is a TVS over $K$ and $f \in L(E; F)$, then $\td f \in L(E; F)$. \item If $\seqi{\rho}$ is a family of seminorms that induces the topology on $E$, then their quotients by $M$ induces the topology on $\td E$. - \end{enumerate} + \end\{enumerate\} + The space $\td E = E/M$ is the \textbf{quotient} of $E$ by $M$. \end{definition} \begin{proof} diff --git a/src/fa/lc/spaces-of-linear.tex b/src/fa/lc/spaces-of-linear.tex index e2fb48c..192c933 100644 --- a/src/fa/lc/spaces-of-linear.tex +++ b/src/fa/lc/spaces-of-linear.tex @@ -6,7 +6,8 @@ Let $T$ be a set, $\sigma \subset 2^T$ be an ideal, $E$ be a locally convex space over $K$, and $\cf \subset E^T$ be a subspace such that \begin{enumerate} \item[(B)] For each $f \in \cf$ and $S \in \sigma$, $f(S) \subset E$ is bounded. - \end{enumerate} + \end\{enumerate\} + For each $S \in \sigma$ and continuous seminorm $\rho: E \to [0, \infty)$, let \[ diff --git a/src/fa/lc/tensor.tex b/src/fa/lc/tensor.tex index f5dd8e3..6bfe890 100644 --- a/src/fa/lc/tensor.tex +++ b/src/fa/lc/tensor.tex @@ -23,7 +23,8 @@ \] is a fundamental system of neighbourhoods at $0$ for $E \otimes_\pi F$. - \end{enumerate} + \end\{enumerate\} + The space $E \otimes_\pi F$ is the \textbf{projective tensor product} of $E$ and $F$, and the mapping $\iota \in L^2(E, F; E \otimes_\pi F)$ is the \textbf{canonical embedding}. @@ -72,7 +73,8 @@ and the seminorm $\rho = p \otimes q$ is the \textbf{cross seminorm} of $p$ and $q$. Moreover, \begin{enumerate} \item[(5)] If the seminorms $\seqi{p}$ define the topology on $E$, and the seminorms $\seqj{q}$ define the topology on $F$, then the seminorms $\bracsn{p_i \otimes q_j| (i, j) \in I \times J}$ define the topology on $E \otimes_\pi F$. - \end{enumerate} + \end\{enumerate\} + \end{definition} \begin{proof}[Proof {{\cite[III.6.3]{SchaeferWolff}}}. ] @@ -140,7 +142,8 @@ \item $\sum_{n \in \natp}|\lambda_n| < \infty$. \item $\limv{n}x_n = 0$ and $\limv{n}y_n = 0$. \item $z = \sum_{n = 1}^\infty \lambda_n x_n \otimes y_n$. - \end{enumerate} + \end\{enumerate\} + \end{theorem} \begin{proof} @@ -162,7 +165,8 @@ \item $v_N = \sum_{k = 1}^{n_N}\lambda_{N, k}x_{N, k} \otimes y_{N, k}$. \item For each $1 \le k \le n_N$, $p_N(x_{N, k}), q_N(x_{N, k}) \le 1/M$. \item $\sum_{k = 1}^{n_N}|\lambda_k| \le 2^{-N+2}$. - \end{enumerate} + \end\{enumerate\} + From here, let $\seqf{(x_j, y_j)} \subset X \times Y$ such that $u_1 = \sum_{j = 1}^n x_j \otimes y_j$, then \[ diff --git a/src/fa/lp/definition.tex b/src/fa/lp/definition.tex index 9a90bd3..009839c 100644 --- a/src/fa/lp/definition.tex +++ b/src/fa/lp/definition.tex @@ -107,7 +107,8 @@ \begin{enumerate} \item[(a)] $f_n \to f$ strongly pointwise. \item[(b)] There exists $g \in L^p(X) \cap L^+(X)$ such that $\norm{f_n}_E \le g$ for all $n \in \natp$. - \end{enumerate} + \end\{enumerate\} + then $f_n \to f$ in $L^p(X; E)$. \end{proposition} diff --git a/src/fa/norm/absolute.tex b/src/fa/norm/absolute.tex index cfb4638..62f0055 100644 --- a/src/fa/norm/absolute.tex +++ b/src/fa/norm/absolute.tex @@ -12,7 +12,8 @@ \begin{enumerate} \item $E$ is a Banach space. \item For any absolutely convergent series $\sum_{n = 1}^\infty x_n$ with $\seq{x_n} \subset E$, there exists $x \in E$ such that $x = \sum_{n = 1}^\infty x_n$. - \end{enumerate} + \end\{enumerate\} + \end{lemma} \begin{proof} @@ -44,7 +45,8 @@ \item If $\sum_{n \in P}x_n = \infty$ but $\sum_{n \in N}x_n > -\infty$, then $\sum_{n = 1}^\infty x_n$ converges to $\infty$ unconditionally. \item If $\sum_{n \in N}x_n = -\infty$ but $\sum_{n \in P}x_n < \infty$, then $\sum_{n = 1}^\infty x_n$ converges to $-\infty$ unconditionally. \item If $\sum_{n \in \natp}|x_n| < \infty$, then $\sum_{n = 1}^\infty x_n$ converges unconditionally. - \end{enumerate} + \end\{enumerate\} + In other words, a series in $\real$ converges unconditionally if and only if its positive parts or its negative parts are finite. \end{theorem} \begin{proof} diff --git a/src/fa/norm/multilinear.tex b/src/fa/norm/multilinear.tex index f2ad6e8..362354f 100644 --- a/src/fa/norm/multilinear.tex +++ b/src/fa/norm/multilinear.tex @@ -8,7 +8,8 @@ \item For each $x \in E$, $y \mapsto T(x, y)$ is a continuous linear map from $F$ to $G$. \item For each $y \in F$, $x \mapsto T(x, y)$ is a continuous linear map from $E$ to $G$. \item $E$ is a Banach space. - \end{enumerate} + \end\{enumerate\} + then $T \in L^2(E, F; G)$. \end{proposition} \begin{proof} diff --git a/src/fa/norm/normed.tex b/src/fa/norm/normed.tex index 717abd6..4b4020d 100644 --- a/src/fa/norm/normed.tex +++ b/src/fa/norm/normed.tex @@ -38,12 +38,14 @@ \begin{enumerate} \item[(a)] $\norm{x}_E \le C\norm{y}_F$. \item[(b)] $\norm{y - Tx}_F \le \gamma \norm{y}_F$. - \end{enumerate} + \end\{enumerate\} + then for any $y \in F$, there exists $\seq{x_n} \subset E$ such that: \begin{enumerate} \item $\sum_{n \in \natp}\norm{x_n}_E \le C\norm{y}_F/(1 - \gamma)$. \item $\sum_{n = 1}^\infty Tx_n = y$. - \end{enumerate} + \end\{enumerate\} + In particular, if $E$ is a Banach space, then for every $y \in F$, there exists $x \in E$ such that $\norm{x}_E \le C\norm{y}_F/(1 - \gamma)$ and $Tx = y$. \end{theorem} \begin{proof} @@ -73,7 +75,8 @@ \begin{enumerate} \item For every $x \in E$, $\sup_{T \in \mathcal{T}}\norm{Tx}_F < \infty$. \item $E$ is a Banach space. - \end{enumerate} + \end\{enumerate\} + then $\sup_{T \in \mathcal{T}}\norm{T}_{L(E; F)} < \infty$. \end{theorem} \begin{proof} diff --git a/src/fa/norm/separable.tex b/src/fa/norm/separable.tex index 8ed9b6b..0c22fb4 100644 --- a/src/fa/norm/separable.tex +++ b/src/fa/norm/separable.tex @@ -29,7 +29,8 @@ \item $\bracs{B(x, r)|x \in E, r > 0}$. \item $\bracsn{\ol{B(x, r)}|x \in E, r > 0}$. \item Open sets in $E$ with respect to the weak topology. - \end{enumerate} + \end\{enumerate\} + \end{proposition} \begin{proof} diff --git a/src/fa/order/lattice.tex b/src/fa/order/lattice.tex index 99462bc..4602a63 100644 --- a/src/fa/order/lattice.tex +++ b/src/fa/order/lattice.tex @@ -7,7 +7,8 @@ \begin{enumerate} \item[(LO1)] For any $x, y, z \in E$ with $x \le y$, $x + z \le y + z$. \item[(LO2)] For any $x, y \in E$ and $\lambda > 0$, $x \le y$ implies that $\lambda x \le \lambda y$. - \end{enumerate} + \end\{enumerate\} + \end{definition} @@ -17,7 +18,8 @@ \begin{enumerate} \item $\sup(A + B) = \sup(A) + \sup(B)$. \item $\sup(A) = -\inf (-A)$ - \end{enumerate} + \end\{enumerate\} + \end{proposition} \begin{proof} @@ -103,12 +105,14 @@ \begin{enumerate} \item[(3)] $|\lambda x| = |\lambda| \cdot |x|$ \item[(4)] $|x + y| \le |x| + |y|$. - \end{enumerate} + \end\{enumerate\} + Finally, for any $x, y \in E$ with $x, y \ge 0$, \begin{enumerate} \item[(5)] $[0, x] + [0, y] = [0, x + y]$. - \end{enumerate} + \end\{enumerate\} + \end{proposition} \begin{proof} @@ -175,7 +179,8 @@ \begin{enumerate} \item For any $x \in C$ and $\lambda \in \real$ with $\lambda \ge 0$, $\phi(\lambda x) = \lambda \phi(x)$. \item For any $x, y \in C$, $\phi(x + y) = \phi(x) + \phi(y)$. - \end{enumerate} + \end\{enumerate\} + then the mapping \[ @@ -218,7 +223,8 @@ \[ |\phi|(x) = \sup\bracs{\phi(y)|y \in E, |y| \le x} \] - \end{enumerate} + \end\{enumerate\} + \end{proposition} \begin{proof} diff --git a/src/fa/rs/bv.tex b/src/fa/rs/bv.tex index adb1fec..3e5292d 100644 --- a/src/fa/rs/bv.tex +++ b/src/fa/rs/bv.tex @@ -67,7 +67,8 @@ \item For each continuous seminorm $\rho$ on $E$, $[\cdot]_{\text{var}, \rho}$ is a seminorm on $BV([a, b]; E)$. \item For each continuous seminorm $\rho$ on $E$, $[\cdot]_{\text{var}, \rho}: E^{[a, b]} \to [0, \infty]$ is lower semicontinuous. In particular, for any $M > 0$, $\bracs{[\cdot]_{\text{var}, \rho} \le M} \subset E^{[a, b]}$ is closed. \item For any $f \in BV([a, b]; E)$ and continuous seminorm $\rho$ on $E$, $\sup_{x \in [a, b]}\rho(f(x)) \le \rho(f(a)) + [f]_{\text{var}, \rho}$. - \end{enumerate} + \end\{enumerate\} + If $(E, \norm{\cdot}_E)$ is a normed vector space, then \begin{enumerate} \item[(5)] $f$ has at most countably many discontinuities. @@ -87,7 +88,8 @@ \begin{enumerate} \item[(a)] $|E_k| \ge N - k$. \item[(b)] $E_k \subset I_k^o$. - \end{enumerate} + \end\{enumerate\} + for $k = 1$. Let $k \le N$ and suppose inductively that $E_k, I_k$ have been constructed. Let $x_k \in E_k$, then by (b), there exists $\eps > 0$ such that $[x_k - \eps, x_k + \eps] \subset I_k$ and $|E_k \setminus [x_k - \eps, x_k + \eps]| \ge N - k$. Let $y_k \in [x_k - \eps, x_k + \eps]$ such that $\norm{f(x_k) - f(y_k)} \ge 1/n$, $I_{k + 1} = I_k \setminus [x_k - \eps, x_k + \eps]$, and $E_{k+1} = E_k \setminus [x_k - \eps, x_k + \eps]$, then $I_k$ and $E_k$ satisfies (a) and (b). diff --git a/src/fa/rs/rs-bv.tex b/src/fa/rs/rs-bv.tex index 05902f8..abe86d2 100644 --- a/src/fa/rs/rs-bv.tex +++ b/src/fa/rs/rs-bv.tex @@ -35,7 +35,8 @@ \begin{enumerate} \item[(a)] For each continuous seminorm $\rho$ on $E$, $[f_\alpha - f]_{u, \rho} \to 0$. \item[(b)] $\lim_{\alpha \in A}\int_a^b f_\alpha dG$ exists. - \end{enumerate} + \end\{enumerate\} + then $f \in RS([a, b], G)$ and $\int_a^b f dG = \lim_{\alpha \in A}\int_a^b f_\alpha dG$. In particular, \begin{enumerate} \item If $H$ is complete, then condition (b) may be omitted. @@ -55,11 +56,13 @@ \begin{enumerate} \item $[f - f_\alpha]_E < \eps/(3[G]_{\text{var}, F})$. \item $\rho\paren{\int_a^b f_\alpha dG - \lim_{\alpha \in A}\int_a^b f_\alpha dG} < \eps/3$. - \end{enumerate} + \end\{enumerate\} + Since $f_\alpha \in RS([a, b], G)$, there exists $P_0 \in \scp([a, b])$ such that if $P \ge P_0$, \begin{enumerate} \item[(3)] $\rho\paren{S(P, c, f_\alpha, G) - \int_a^b f_\alpha dG} < \eps/3$. - \end{enumerate} + \end\{enumerate\} + Thus for any $(P, c) \in \scp_t([a, b])$ with $P \ge P_0$, \[ \rho\paren{S(P, c, f, G) - \lim_{\alpha \in A}\int_a^b f_\alpha dG} < \eps diff --git a/src/fa/tvs/complete-metric.tex b/src/fa/tvs/complete-metric.tex index ced164c..11bef0b 100644 --- a/src/fa/tvs/complete-metric.tex +++ b/src/fa/tvs/complete-metric.tex @@ -21,12 +21,14 @@ \begin{enumerate} \item[(a)] $\eta(y - Tx) \le \gamma \eta(y)$. \item[(b)] $\rho(x) \le C \eta(y)$. - \end{enumerate} + \end\{enumerate\} + then for any $y \in F$, there exists $\seq{x_n} \subset E$ such that: \begin{enumerate} \item $\sum_{n \in \natp}\rho(x_n) \le C\eta(y)/(1 - \gamma)$. \item $y = \limv{N}\sum_{n = 1}^N Tx_n$. - \end{enumerate} + \end\{enumerate\} + In particular, \[ T\braks{B_E\paren{0, \frac{Cr}{(1 - \gamma)}}} \supset B_F(0, r) @@ -38,12 +40,14 @@ \begin{enumerate} \item[(I)] $\sum_{n = 1}^N\rho(x_n) \le C\eta(y)\sum_{n = 0}^{N-1}\gamma^{n}$. \item[(II)] $\eta\paren{y - \sum_{n = 1}^N Tx_n} \le \eta(y)\gamma^N$. - \end{enumerate} + \end\{enumerate\} + By assumption, there exists $x_{N+1} \in E$ such that: \begin{enumerate} \item[(i)] $\eta\paren{y - \sum_{n = 1}^{N+1} Tx_n} \le \gamma \eta\paren{y - \sum_{n = 1}^N Tx_n} \le \gamma^{N+1}$. \item[(ii)] $\rho(x_{N+1}) \le C\eta\paren{y - \sum_{n = 1}^N Tx_n} \le C\eta(y)\gamma^N$. - \end{enumerate} + \end\{enumerate\} + Combining (I) and (ii) shows that $\sum_{n = 1}^N \rho(x_n) \le C \eta(y) \sum_{n = 0}^N \gamma^n$. Therefore there exists $\seq{x_n} \subset E$ such that (I) and (II) holds for all $N \in \natp$. By (I), $\sum_{n \in \natp}\rho(x_n) \le C\eta(y)\sum_{n \in \natz}\gamma^n = C \eta(y)/(1 - \gamma)$. By (II), $\limv{N}\eta\paren{y - \limv{N}\sum_{n = 1}^N Tx_n} = \limv{N}\eta(y)\gamma^N = 0$. @@ -55,7 +59,8 @@ \begin{enumerate} \item[(a)] For any $r > 0$, there exists $\delta(r) > 0$ such that $\overline{T(B_E(0, r))} \supset B_F(0, \delta(r))$. \item[(b)] $E$ is complete. - \end{enumerate} + \end\{enumerate\} + then for every $s > r$, $T(B_E(0, s)) \supset B_F(0, \delta(r))$. \end{proposition} \begin{proof} @@ -65,12 +70,14 @@ \item[(ii)] $s_1 = r$. \item[(iii)] For all $n \in \natp$, $\overline{T(B_E(0, s_n))} \supset B_F(0, \delta_n)$. \item[(iv)] $\rho_1 = \rho$. - \end{enumerate} + \end\{enumerate\} + Let $y_0 \in B(0, r)$ and $x_0 = 0$. Let $N \in \natp$ and suppose inductively that $\bracs{x_n}_1^N \subset E$ has been constructed such that: \begin{enumerate} \item[(I)] For each $0 \le n \le N - 1$, $\rho(x_{n+1} - x_n) < s_n$. \item[(II)] For each $0 \le n \le N$, $\eta(Tx_n - y) \le \rho_{n+1}$. - \end{enumerate} + \end\{enumerate\} + By density of $T(x_N + B_E(0, s_N))$ in $Tx_N + B_F(0, \rho_N)$, there exists $x_{N+1} \in T(x_N + B_E(0, s_N))$ such that $\eta(Tx_{N+1} - y) \le \rho_{N+2}$. By (I), $\seq{x_N}$ is a Cauchy sequence, so @@ -88,7 +95,8 @@ \begin{enumerate} \item[(a)] For any $r > 0$, there exists $C \ge 0$ such that for any $y \in T(E)$, there exits $x \in T^{-1}(y)$ with $\rho(x) \le C\eta(y)$. \item[(b)] $E$ is complete. - \end{enumerate} + \end\{enumerate\} + then $T(E)$ is closed. \end{proposition} \begin{proof} diff --git a/src/fa/tvs/completion.tex b/src/fa/tvs/completion.tex index d5141d4..169d25b 100644 --- a/src/fa/tvs/completion.tex +++ b/src/fa/tvs/completion.tex @@ -8,11 +8,13 @@ \item $\wh E$ is a complete separated TVS. \item $\iota \in L(E; \wh E)$. \item[(U)] For any $(F, T)$ satisfying (1) and (2), there exists a unique $\ol{T} \in L(\wh E; F)$ such that the following diagram commutes: - \end{enumerate} + \end\{enumerate\} + Moreover, \begin{enumerate} \item[(4)] $\iota(E)$ is dense in $\wh E$. - \end{enumerate} + \end\{enumerate\} + The pair $(\wh E, \iota)$ is the \textbf{Hausdorff completion} of $E$. \end{definition} \begin{proof} diff --git a/src/fa/tvs/continuous.tex b/src/fa/tvs/continuous.tex index 13fe2e4..88a87f5 100644 --- a/src/fa/tvs/continuous.tex +++ b/src/fa/tvs/continuous.tex @@ -9,7 +9,8 @@ \item $T \in UC(E; F)$. \item $T \in C(E; F)$. \item $T$ is continuous at $0$. - \end{enumerate} + \end\{enumerate\} + If the above holds, then $T$ is a \textbf{continuous linear map}. The set $L(E; F)$ denotes the vector space of all continuous linear maps from $E$ to $F$. \end{definition} \begin{proof} @@ -48,7 +49,8 @@ } \] - \end{enumerate} + \end\{enumerate\} + The uniformity $\fU$ and its induced topology are the \textbf{product uniformity/topology}, and $E$ equipped with $\fU$ is the \textbf{product TVS} of $\seqi{E}$. \end{definition} diff --git a/src/fa/tvs/definition.tex b/src/fa/tvs/definition.tex index ef6e95a..caa0dde 100644 --- a/src/fa/tvs/definition.tex +++ b/src/fa/tvs/definition.tex @@ -8,7 +8,8 @@ \begin{enumerate} \item[(TVS1)] $E \times E \to E$ with $(x, y) \mapsto x + y$ is continuous. \item[(TVS2)] $K \times E \to E$ with $(\lambda, x) \mapsto \lambda x$ is continuous. - \end{enumerate} + \end\{enumerate\} + then the pair $(E, \topo)$ is a \textbf{topological vector space}. \end{definition} @@ -50,7 +51,8 @@ \begin{enumerate} \item There exists a unique translation-invariant uniformity $\fU$ on $E$ that induces the topology on $E$. \item For each neighbourhood $V \in \cn(0)$, let $U_V = \bracs{(x, y) \in E^2| x - y \in V}$, then for any fundamental system of neighbourhoods $\fB_0$ at $0$, $\fB = \bracs{U_V| V \in \fB_0}$ is a fundamental system of entourages for $\fU$. - \end{enumerate} + \end\{enumerate\} + The space $E$ will always be assumed to be equipped with its translation-invariant uniformity. \end{proposition} \begin{proof} @@ -162,12 +164,14 @@ \begin{enumerate} \item[(TVB1)] For each $U \in \fB$, there exists $V \in \fB$ such that $V + V \subset U$. \item[(TVB2)] For each $U \in \fB$, $U$ is circled and radial. - \end{enumerate} + \end\{enumerate\} + Conversely, if $\fB \subset 2^E$ is a family of sets that contain $0$ and satisfies (TVB1) and (TVB2), then there exists a unique topology $\topo$ on $E$ such that: \begin{enumerate} \item $\topo$ is translation-invariant. \item $\fB$ is a fundamental system of neighbourhoods at $0$ for $\topo$. - \end{enumerate} + \end\{enumerate\} + Moreover, \begin{enumerate} \item[(3)] $(E, \topo)$ is a TVS. @@ -186,7 +190,8 @@ \item[(FB1)] For any $V, V' \in \fB$, there exists $W \in \fB$ with $W \subset V \cap V'$. In which case, $U_{V} \cap U_{V'} \supset U_W \in \mathfrak{V}$. \item[(UB1)] For any $x \in E$ and $V \in \fB$, $x - x = 0 \in V$, so $\Delta \subset U_V$. \item[(UB2)] For any $V \in \fB$, by (TVB1), there exists $W \in \fB$ such that $W + W \subset V$. In which case, for any $x, y, z \in E$ with $x - y, y - z \in W$, $x - z \in V$. Therefore $U_W \circ U_W \subset U_V$. - \end{enumerate} + \end\{enumerate\} + By \autoref{proposition:fundamental-entourage-criterion}, there exists a unique uniformity $\fU$ on $E$ for which $\mathfrak{V}$ is a fundamental system of entourages. (1): Since $\mathfrak{V}$ is translation-invariant, so is $\fU$. diff --git a/src/fa/tvs/dual.tex b/src/fa/tvs/dual.tex index b7c59a1..45a9b07 100644 --- a/src/fa/tvs/dual.tex +++ b/src/fa/tvs/dual.tex @@ -7,7 +7,8 @@ \begin{enumerate} \item $u \in \hom(E; \real)$ when $E$ is viewed as a vector space over $\real$. \item For any $x \in E$, $\dpb{x, \phi}{E} = \dpb{x, u}{E} - i \dpb{ix, u}{E}$. - \end{enumerate} + \end\{enumerate\} + Conversely, if $u \in \hom(E; \real)$ and $\phi \in \hom(E; \complex)$ is defined by $\dpb{x, \phi}{E} = \dpb{x, u}{E} - i \dpb{ix, u}{E}$ for all $x \in E$, then $f \in \hom(E; \complex)$. \end{proposition} \begin{proof}[Proof {{\cite[Proposition 5.5]{Folland}}}. ] diff --git a/src/fa/tvs/inductive.tex b/src/fa/tvs/inductive.tex index a5cb0b3..dd8bb9b 100644 --- a/src/fa/tvs/inductive.tex +++ b/src/fa/tvs/inductive.tex @@ -9,7 +9,8 @@ \item For each $i \in I$, $T_i \in L(E_i; E)$. \item[(U)] For any topology $\mathcal{S}$ on $E$ satisfying (1) and (2), $\mathcal{S} \subset T$. \item For any TVS $F$ and $T \in \hom(E; F)$, $T \in L(E; F)$ if and only if $T \circ T_i \in L(E_i; F)$ for all $i \in I$. - \end{enumerate} + \end\{enumerate\} + The topology $\topo$ is the \textbf{inductive topology} on $E$ induced by $\seqi{T}$. \end{definition} \begin{proof} @@ -61,7 +62,8 @@ for all $i \in I$. \item For any TVS $F$ and $T \in \hom(E; F)$, $T \in L(E; F)$ if and only if $T \circ T^i_E \in L(E_i; F)$ for all $i \in I$. - \end{enumerate} + \end\{enumerate\} + The pair $(E, \bracsn{T^i_E}_{i \in I})$ is the \textbf{inductive limit} of $(\seqi{E}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$. \end{definition} \begin{proof} diff --git a/src/fa/tvs/metric.tex b/src/fa/tvs/metric.tex index 5a1e1e1..b3a381e 100644 --- a/src/fa/tvs/metric.tex +++ b/src/fa/tvs/metric.tex @@ -82,7 +82,8 @@ \begin{enumerate} \item[(a)] For each $n \in \natp$, $U_n$ is circled, radial, and contains $0$. \item[(b)] For each $n \in \natp$, $U_{n+1} + U_{n+1} \subset U_n$. - \end{enumerate} + \end\{enumerate\} + then there exists a pseudonorm $\rho: E \to [0, \infty)$ such that for each $n \in \natp$, \[ U_{n+1} \subset \rho^{-1}([0, 2^{-n})) \subset U_{n} @@ -110,7 +111,8 @@ so $\rho(\lambda x) \le \rho(x)$. \item[(PN3)] Let $x, y \in X$ and $M, N \subset \natp$ finite such that $x \in U_M$ and $y \in U_N$. Assume without loss of generality that $\rho_M + \rho_N < 1$, then there exists a unique $P \subset \nat$ finite such that $\rho_P = \rho_M + \rho_N$. In which case, $U_P \supset U_M + U_N$ by assumption (b). Therefore $\rho(x + y) \le \rho(x) + \rho(y)$. - \end{enumerate} + \end\{enumerate\} + For any $x \in U_{n+1}$, $\rho(x) \le 2^{-n+1} < 2^n$, so $U_{n+1} \subset \rho^{-1}([0, 2^{-n}))$ by \autoref{proposition:dyadic-semigroup-order}. On the other hand, for any $x \in E$ with $\rho(x) < 2^{-n}$, $x \in U_{2^{-n}} = U_n$. This allows showing the remaining seminorm axioms by considering neighbourhoods of the form $\bracs{U_n|n \in \natp}$. \begin{enumerate} \item[(PN4)] Let $x \in X$ and $n \in \natp$. By assumption (a), there exists $\alpha > 0$ such that for any $\lambda \in K$ with $\abs{\lambda} \ge \alpha$, $x \in \lambda U_n$. Therefore for any $\lambda \in K$ with $\abs{\lambda} \le \alpha^{-1}$, $\lambda x \in U_n$, and $\rho(x) \le 2^{-n}$. diff --git a/src/fa/tvs/projective.tex b/src/fa/tvs/projective.tex index 98c5405..9ee0717 100644 --- a/src/fa/tvs/projective.tex +++ b/src/fa/tvs/projective.tex @@ -7,7 +7,8 @@ \begin{enumerate} \item For each $i \in I$, $T_i \in L(E; F_i)$. \item[(U)] If $\mathfrak{V}$ is a uniformity on $E$ satisfying $(1)$, then $\mathfrak{V} \supset \fU$. - \end{enumerate} + \end\{enumerate\} + Moreover, \begin{enumerate} \item[(3)] $\fU$ is translation-invariant. @@ -19,7 +20,8 @@ \] is a fundamental system of neighbourhoods for $E$ at $0$. - \end{enumerate} + \end\{enumerate\} + The uniformity $\fU$ and its topology are the \textbf{projective uniformity/topology} induced by $\seqi{T}$. \end{definition} @@ -74,7 +76,8 @@ for all $i \in I$. \item For any TVS $F$ over $K$ and $S \in \hom(F; E)$, $S \in L(F; E)$ if and only if $T^E_i \circ S \in L(F; E_i)$ for all $i \in I$. - \end{enumerate} + \end\{enumerate\} + The pair $(E, \bracsn{T^E_i}_{i \in I})$ is the \textbf{projective limit} of $(\seqi{E}, \bracsn{T^i_j|i, j \in I, i \lesssim j})$. \end{definition} \begin{proof} diff --git a/src/fa/tvs/quotient.tex b/src/fa/tvs/quotient.tex index 38632a0..db8db20 100644 --- a/src/fa/tvs/quotient.tex +++ b/src/fa/tvs/quotient.tex @@ -17,7 +17,8 @@ \] If $F$ is a TVS over $K$ and $f \in L(E; F)$, then $\td f \in L(E; F)$. - \end{enumerate} + \end\{enumerate\} + The space $\td E = E/M$ is the \textbf{quotient} of $E$ by $M$. \end{definition} \begin{proof} @@ -32,7 +33,8 @@ \begin{enumerate} \item[(TVB1)] Let $U \in \cn(0)$ be circled and radial. For any $\lambda \in K$ with $\abs{\lambda} \le 1$, $\lambda \pi(U) = \pi(\lambda U) \subset \pi(U)$, so $\pi(U)$ is also circled. For any $x + M \in E/M$, there exists $\lambda \in K$ such that $x \in \lambda U$. In which case, $x \in \lambda U + M = \pi(U)$, so $\pi(U)$ is also radial. \item[(TVB2)] For any $U \in \cn(0)$ circled and radial, by \autoref{proposition:tvs-good-neighbourhood-base}, there exists $W \in \cn(0)$ such that $W + W \subset U$. In which case, $\pi(W) + \pi(W) \subset \pi(U)$. - \end{enumerate} + \end\{enumerate\} + By \autoref{proposition:tvs-0-neighbourhood-base}, there exists a unique translation-invariant topology on $E/M$ such that $\fB$ is a fundamental system of neighbourhoods at $0$, which must be the quotient topology on $E/M$. In which case, the quotient topology is a vector space topology by (3) of \autoref{proposition:tvs-0-neighbourhood-base}. (2), (3), (U): By \autoref{definition:quotient-topology}. diff --git a/src/fa/tvs/spaces-of-linear.tex b/src/fa/tvs/spaces-of-linear.tex index 270a83f..c412e1c 100644 --- a/src/fa/tvs/spaces-of-linear.tex +++ b/src/fa/tvs/spaces-of-linear.tex @@ -115,7 +115,8 @@ \] is an isomorphism. - \end{enumerate} + \end\{enumerate\} + which allows the identification \[ \underbrace{B_{\sigma}(E; B_{\sigma}(E; \cdots)))}_{k \text{ times}} = B^k_{\sigma}(E; F) @@ -166,7 +167,8 @@ \begin{enumerate} \item[(a)] There exists a dense subset $S \subset E$ such that $T_\alpha x \to Tx$ strongly for all $x \in S$. \item[(b)] $\bracs{T_\alpha|\alpha \in A}$ is uniformly equicontinuous. - \end{enumerate} + \end\{enumerate\} + then $T_\alpha \to T$ in $L_s(E; F)$. \end{proposition} diff --git a/src/measure/bochner-integral/bochner.tex b/src/measure/bochner-integral/bochner.tex index e5e5d46..a19e0da 100644 --- a/src/measure/bochner-integral/bochner.tex +++ b/src/measure/bochner-integral/bochner.tex @@ -52,7 +52,8 @@ \begin{enumerate} \item[(a)] $f_n \to f$ strongly pointwise. \item[(b)] There exists $g \in L^1(X) \cap L^+(X)$ such that $\norm{f_n}_E \le g$ for all $n \in \natp$. - \end{enumerate} + \end\{enumerate\} + then $\int f d\mu = \limv{n}\int f_n d\mu$. \end{theorem} @@ -76,7 +77,8 @@ \begin{enumerate} \item For any $x \in E$ and $A \in \cm$, $I_\lambda(x \cdot \one_A) = x \mu(A)$. \item For any $f \in L^1(X, |\mu|; E)$, $\normn{I_\lambda f}_{G} \le \norm{\lambda}_{L^2(E, F; G)} \cdot \norm{f}_{L^1(X, |\mu|; E)}$. - \end{enumerate} + \end\{enumerate\} + For any $f \in L^1(X; E)$, $I_\lambda f = \int \lambda(f, d\mu)$ is the \textbf{Bochner integral} of $f$ with respect to $\mu$ and $\lambda$. \end{definition} @@ -96,7 +98,8 @@ \begin{enumerate} \item[(a)] $f_n \to f$ strongly pointwise. \item[(b)] There exists $g \in L^1(X) \cap L^+(X)$ such that $\norm{f_n}_E \le g$ for all $n \in \natp$. - \end{enumerate} + \end\{enumerate\} + then $\int \lambda(f , d\mu) = \limv{n}\int \lambda(f_n, d\mu)$. \end{theorem} diff --git a/src/measure/bochner-integral/strongly.tex b/src/measure/bochner-integral/strongly.tex index d3a569d..add2d3d 100644 --- a/src/measure/bochner-integral/strongly.tex +++ b/src/measure/bochner-integral/strongly.tex @@ -11,7 +11,8 @@ \begin{enumerate} \item[(a)] For each $n \in \natp$, $\norm{f_n}_E \le \norm{f}_E$. \item[(b)] $\norm{f_n(x) - f(x)}_E \to 0$ pointwise as $n \to \infty$. - \end{enumerate} + \end\{enumerate\} + \end{enumerate} If the above holds, then $f$ is a \textbf{strongly measurable} function. @@ -39,7 +40,8 @@ \begin{enumerate} \item For any strongly measurable functions $f, g: X \to E$ and $\lambda \in K$, $\lambda f + g$ is strongly measurable. \item For any strongly measurable functions $\bracs{f_n: X \to E|n \in \natp}$ and $f: X \to E$, if $f_n \to f$ strongly pointwise, then $f$ is strongly measurable. - \end{enumerate} + \end\{enumerate\} + \end{proposition} \begin{proof} diff --git a/src/measure/lebesgue-integral/complex.tex b/src/measure/lebesgue-integral/complex.tex index 7b2c494..3f7f80b 100644 --- a/src/measure/lebesgue-integral/complex.tex +++ b/src/measure/lebesgue-integral/complex.tex @@ -104,7 +104,8 @@ \begin{enumerate} \item $f_n \to f$ pointwise. \item There exists $g \in \mathcal{L}^1(X)$ such that $\abs{f_n} \le \abs{g}$ for all $n \in \natp$. - \end{enumerate} + \end\{enumerate\} + then $\int fd\mu = \limv{n}\int f_n d\mu$. \end{theorem} \begin{proof}[Proof {{\cite[Theorem 2.24]{Folland}}}. ] diff --git a/src/measure/measurable-maps/metric.tex b/src/measure/measurable-maps/metric.tex index 87cec8e..840f1c1 100644 --- a/src/measure/measurable-maps/metric.tex +++ b/src/measure/measurable-maps/metric.tex @@ -76,7 +76,8 @@ \item[(a)] For each $y \in Y$, $y \in \ol{N(y)^o}$. \item[(b)] $\bigcap_{y \in Y}N(y) \ne \emptyset$. \item[(c)] For any $y_0 \in Y$, $\bracs{y \in Y|y_0 \in N(y)} \in \cb_Y$. - \end{enumerate} + \end\{enumerate\} + Then, for any $f: X \to Y$, the following are equivalent: \begin{enumerate} \item $f$ is $(\cm, \cb_Y)$-measurable. @@ -84,7 +85,8 @@ \begin{enumerate} \item[(i)] For each $x \in X$ and $n \in \natp$, $f_n(x) \in N(f(x))$. \item[(ii)] $f_n \to f$ pointwise. - \end{enumerate} + \end\{enumerate\} + \item There exists a sequence $\seq{f_n}$ of $(\cm, \cb_Y)$-measurable simple functions such that $f_n \to f$ pointwise. \end{enumerate} \end{proposition} @@ -146,6 +148,7 @@ \bracs{y \in E|y_0 \in N(y)} = \bracs{y \in E|\norm{y_0}_E \le \norm{y}_E} \in \cb_E \] - \end{enumerate} + \end\{enumerate\} + By \autoref{proposition:measurable-simple-separable}, (1) and (2) are equivalent. \end{proof} diff --git a/src/measure/measurable-maps/real-valued.tex b/src/measure/measurable-maps/real-valued.tex index 80a0eed..cea499b 100644 --- a/src/measure/measurable-maps/real-valued.tex +++ b/src/measure/measurable-maps/real-valued.tex @@ -24,7 +24,8 @@ \item $G = \limsup_{n \to \infty}f_n$. \item $g = \limsup_{n \to \infty}f_n$. \item $\limv{n}f_n$ (if it exists). - \end{enumerate} + \end\{enumerate\} + In addition, if the above functions are $\real$-valued, then they are $(\cm, \cb_{\real})$-measurable. \end{proposition} \begin{proof} diff --git a/src/measure/measure/complete.tex b/src/measure/measure/complete.tex index 937eae3..633a7e0 100644 --- a/src/measure/measure/complete.tex +++ b/src/measure/measure/complete.tex @@ -20,7 +20,8 @@ \item There exists an extension $\ol{\mu}$ of $\mu$ as a measure on $\cm$. \item $(X, \ol{\cm}, \ol{\mu})$ is a complete measure space. \item[(U)] For any complete measure space $(X, \cf, \nu)$ where $\cf \supset \cm$ and $\nu|_\cm = \mu$, $\cf \supset \ol{\cm}$ and $\nu|_{\ol{\cm}} = \ol{\mu}$. - \end{enumerate} + \end\{enumerate\} + and space $(X, \ol{\cm}, \mu)$ is the \textbf{completion} of $(X, \cm, \mu)$. \end{definition} \begin{proof} diff --git a/src/measure/measure/kolmogorov.tex b/src/measure/measure/kolmogorov.tex index aaa4119..9e35a8a 100644 --- a/src/measure/measure/kolmogorov.tex +++ b/src/measure/measure/kolmogorov.tex @@ -16,13 +16,15 @@ \item[(a)] Every finite measure on $\prod_{j = 1}^n X_j$ is regular. \item[(b)] $X_n$ is Hausdorff. \item[(c)] $X_n$ is separable. - \end{enumerate} + \end\{enumerate\} + Let $\bracs{\mu_{I}| I \subset \natp \text{ finite}}$ be consistent Borel probability measures, then for any $\seq{B_n}$ where: \begin{enumerate} \item[(d)] For each $n \in \nat$, $B_n \in \cb_{\prod_{j = 1}^n X_j}$. \item[(e)] For each $n \in \nat$, $B_{n+1} \subset B_n \times X_{n+1}$. \item[(f)] There exists $\eps > 0$ such that $\mu_{[n]}(B_n) > \eps$ for all $n \in \natp$. - \end{enumerate} + \end\{enumerate\} + Then there exists $\seq{K_n}$ such that for every $n \in \natp$, \begin{enumerate} \item $K_n \subset \prod_{j = 1}^n X_j$ is compact. @@ -52,7 +54,8 @@ \item[(a)] Every finite measure on $\prod_{j \in J} X_j$ is regular. \item[(b)] $X_j$ is Hausdorff. \item[(c)] $X_j$ is separable. - \end{enumerate} + \end\{enumerate\} + Let $\bracs{\mu_{I}| I \subset \natp \text{ finite}}$ be consistent Borel probability measures, then there exists a unique probability measure $\mu: \bigotimes_{i \in I}\cb_{X_i} \to [0, 1]$ such that for any $J \subset I$ finite, $\mu = \mu_J \circ \pi_J^{-1}$. \end{theorem} \begin{proof}[Proof {{\cite[Theorem 1.14]{Baudoin}}}. ] @@ -71,7 +74,8 @@ \item $K_n \subset B_n$. \item $K_{n+1} \subset K_n \times X_{n+1}$. \item $\mu(K_n) \ge \eps/2$. - \end{enumerate} + \end\{enumerate\} + Let $N \in \natp$ and $x \in \prod_{j = 1}^N X_j$ such that $x \in \bigcap_{n \ge N}\pi_{[N]}(K_n)$. By compactness and (4), there exists $x_{N+1} \in X_{N+1}$ such that $(x_1, \cdots, x_N, x_{N+1}) \in \bigcap_{n > N}\pi_{[N+1]}(K_n)$. Thus there exists $x \in \prod_{i \in I}X_i$ such that $x \in \pi_{[n]}^{-1}(K_n)$ for all $n \in \natp$, and \[ x \in \bigcap_{n \in \natp}\pi_{[n]}^{-1}(K_n) \subset \bigcap_{n \in \natp}\pi_{[n]}^{-1}(B_n) \ne \emptyset diff --git a/src/measure/measure/lebesgue-stieltjes.tex b/src/measure/measure/lebesgue-stieltjes.tex index f0428f8..c33169d 100644 --- a/src/measure/measure/lebesgue-stieltjes.tex +++ b/src/measure/measure/lebesgue-stieltjes.tex @@ -44,7 +44,8 @@ \] \item[(U)] For any function $G: \real \to \real$ satisfying (1), $F - G$ is constant. - \end{enumerate} + \end\{enumerate\} + Conversely, if $F: \real \to \real$ is a Stieltjes function, then there exists a unique Borel measure $\mu_F: \cb_\real \to [0, \infty]$ satisfying (1), and $\mu_F$ is the \textbf{Lebesgue-Stieltjes measure} associated with $F$. \end{definition} \begin{proof}[Proof {{\cite[Theorem 1.16]{Folland}}}. ] diff --git a/src/measure/measure/measure.tex b/src/measure/measure/measure.tex index 095a371..9634cd9 100644 --- a/src/measure/measure/measure.tex +++ b/src/measure/measure/measure.tex @@ -14,13 +14,15 @@ \begin{enumerate} \item[(M1)] $\mu(\emptyset) = 0$. \item[(M2)] For any $\seq{E_n} \subset \cm$ pairwise disjoint, $\mu\paren{\bigsqcup_{n \in \natp}E_n} = \sum_{n \in \natp} \mu(E_n)$. - \end{enumerate} + \end\{enumerate\} + In which case, $(X, \cm, \mu)$ is a \textbf{measure space}. If $\mu: \cm \to [0, \infty]$ instead satisfies (M1) and \begin{enumerate} \item[(M2')] For any $\seqf{E_j} \subset \cm$ pairwise disjoint, $\mu\paren{\bigsqcup_{j = 1}^n E_j} = \sum_{j = 1}^n \mu(E_j)$. - \end{enumerate} + \end\{enumerate\} + then $\mu$ is a \textbf{finitely-additive measure}. \end{definition} @@ -100,7 +102,8 @@ \item[(a)] $\sigma(\mathcal{P}) = \cm$. \item[(b)] $\mu(E) = \nu(E)$ for all $E \in \mathcal{P}$. \item[(c)] There exists $\seq{E_n} \subset \mathcal{P}$ such that $E_n \upto X$ and $\mu(E_n) < \infty$ for all $n \in \natp$. - \end{enumerate} + \end\{enumerate\} + then $\mu = \nu$. \end{theorem} \begin{proof} @@ -122,7 +125,8 @@ \] by continuity from below (\autoref{proposition:measure-properties}). - \end{enumerate} + \end\{enumerate\} + so $\alg(F)$ is a $\lambda$-system. By (a) and Dynkin's $\pi$-$\lambda$ theorem (\autoref{theorem:pi-lambda}), $\alg(F) = \cm$. Let $\seq{E_n}$ as in assumption (c), then $\mu(E_n \cap F) = \mu(E_n \cap F)$ for all $n \in \natp$ and $F \in \cm$. Thus by continuity from below (\autoref{proposition:measure-properties}), @@ -163,7 +167,8 @@ \[ \int g df_*\mu = \int g \circ f d\mu \] - \end{enumerate} + \end\{enumerate\} + \end{definition} \begin{proof} diff --git a/src/measure/measure/outer.tex b/src/measure/measure/outer.tex index 9c9be51..b33a37c 100644 --- a/src/measure/measure/outer.tex +++ b/src/measure/measure/outer.tex @@ -115,7 +115,8 @@ \item[(a)] $\nu|_{\cm \cap \cn} \le \mu_{\cm \cap \cn}$. \item[(b)] For any $E \in \cm \cap \cn$ with $\mu(E) < \infty$, $\mu(E) = \nu(E)$. \item[(c)] If $\mu$ is $\sigma$-finite, then $\nu = \mu$. - \end{enumerate} + \end\{enumerate\} + \end{enumerate} \end{theorem} \begin{proof}[Proof {{\cite[Theorem 1.14]{Folland}}}. ] diff --git a/src/measure/measure/product.tex b/src/measure/measure/product.tex index 97702f0..06ccb76 100644 --- a/src/measure/measure/product.tex +++ b/src/measure/measure/product.tex @@ -8,7 +8,8 @@ \begin{enumerate} \item For each $E \in \cm$ and $F \in \cn$, $\mu \otimes \nu(E \times F) = \mu(E)\nu(F)$. \item[(U)] For any measure $\lambda: \cm \otimes \cn \to [0, \infty]$, $\lambda \le \mu$. For any $A \in \cm \otimes \cn$ with $\mu(A) < \infty$, $\lambda(A) = \mu(A)$. In particular, if $\mu$ is $\sigma$-finite, then $\lambda = \mu$. - \end{enumerate} + \end\{enumerate\} + The measure $\mu \otimes \nu$ is the \textbf{product} of $\mu$ and $\nu$. \end{definition} @@ -57,7 +58,8 @@ \begin{enumerate} \item For any $E \in \cm \otimes \cn$, $x \in X$, and $y \in Y$, $\bracs{z \in Y|(x, z) \in E} \in \cm$ and $\bracs{z \in X|(z, y) \in E} \in \cn$. \item For any measure space $(Z, \cf)$, $(\cm \otimes \cn, \cf)$-measurable function $f: X \times Y \to Z$, $x \in X$, and $y \in Y$, $f(x, \cdot)$ is $(\cn, \cf)$-measurable and $f(\cdot, y)$ is $(\cm, \cf)$-measurable. - \end{enumerate} + \end\{enumerate\} + \end{lemma} \begin{proof} @@ -100,8 +102,10 @@ \int_{X \times Y}f(z)\mu \otimes \nu(dz) &= \int_X\int_Y f(x, y)\nu(dy)\mu(dx) \\ &= \int_{Y}\int_X f(x, y)\mu(dx)\nu(dy) \end{align*} - \end{enumerate} - \end{enumerate} + \end\{enumerate\} + + \end\{enumerate\} + \end{theorem} \begin{proof} diff --git a/src/measure/measure/regular.tex b/src/measure/measure/regular.tex index 747543d..5c17859 100644 --- a/src/measure/measure/regular.tex +++ b/src/measure/measure/regular.tex @@ -31,6 +31,7 @@ \item[(a)] $X$ is a LCH space. \item[(b)] Every open set of $X$ is $\sigma$-compact. \item[(c)] For any $K \subset X$ compact, $\mu(K) < \infty$. - \end{enumerate} + \end\{enumerate\} + then $\mu$ is a regular measure. \end{theorem} diff --git a/src/measure/measure/semifinite.tex b/src/measure/measure/semifinite.tex index 6db4b11..756f06c 100644 --- a/src/measure/measure/semifinite.tex +++ b/src/measure/measure/semifinite.tex @@ -11,7 +11,8 @@ \mu(E) = \sup\bracs{\mu(F)| F \in \cm, F \subset E, \mu(F) < \infty} \] - \end{enumerate} + \end\{enumerate\} + If the above holds, then $\mu$ is a \textbf{semifinite measure}. \end{definition} \begin{proof} diff --git a/src/measure/measure/sigma-finite.tex b/src/measure/measure/sigma-finite.tex index f37d632..9a6dd4b 100644 --- a/src/measure/measure/sigma-finite.tex +++ b/src/measure/measure/sigma-finite.tex @@ -7,6 +7,7 @@ \begin{enumerate} \item There exists $\seq{E_n} \subset \cm$ pairwise disjoint such that $\bigsqcup_{n \in \nat}E_n = X$ and $\mu(E_n) < \infty$ for all $n \in \nat$. \item There exists $\seq{E_n} \subset \cm$ such that $E_n \upto X$ and $\mu(E_n) < \infty$ for all $n \in \nat$. - \end{enumerate} + \end\{enumerate\} + If the above holds, then $\mu$ is a \textbf{$\sigma$-finite measure}. \end{definition} diff --git a/src/measure/radon/c0.tex b/src/measure/radon/c0.tex index 2c3d217..60934d0 100644 --- a/src/measure/radon/c0.tex +++ b/src/measure/radon/c0.tex @@ -8,7 +8,8 @@ \item $I = I^+ - I^-$. \item $I^+ \perp I^-$. \item $\norm{I^+}_{C_0(X; \real)^*}, \norm{I^-}_{C_0(X; \real)^*} \le \norm{I}_{C_0(X; \real)^*}$. - \end{enumerate} + \end\{enumerate\} + \end{lemma} \begin{proof} diff --git a/src/measure/radon/radon.tex b/src/measure/radon/radon.tex index 6cd9d17..6f57ae1 100644 --- a/src/measure/radon/radon.tex +++ b/src/measure/radon/radon.tex @@ -115,7 +115,8 @@ \begin{enumerate} \item[(a)] For any $U \subset X$ open, $U$ is $\sigma$-compact. \item[(b)] For any $K \subset X$ compact, $\mu(K) < \infty$. - \end{enumerate} + \end\{enumerate\} + then $\mu$ is a regular measure on $X$. \end{proposition} \begin{proof} diff --git a/src/measure/radon/riesz.tex b/src/measure/radon/riesz.tex index 6d9a1ba..0efa3a6 100644 --- a/src/measure/radon/riesz.tex +++ b/src/measure/radon/riesz.tex @@ -71,7 +71,8 @@ \] As this holds for all $\eps > 0$, $\mu^*(E) \le \sum_{n \in \natp}\mu^*(E_n)$. - \end{enumerate} + \end\{enumerate\} + Therefore $\mu^*: 2^E \to [0, \infty]$ is an outer measure. To see that all Borel sets are $\mu^*$-measurable, let $E \subset X$ and $U \in \topo$. First suppose that $E$ is open. Let $f \prec E \cap U$, then for any $g \prec E \setminus \supp{f}$, $\supp{f} \cap \supp{g} = \emptyset$ and $f + g \prec E$, so diff --git a/src/measure/sets/algebra.tex b/src/measure/sets/algebra.tex index 592a552..e724856 100644 --- a/src/measure/sets/algebra.tex +++ b/src/measure/sets/algebra.tex @@ -38,7 +38,8 @@ \begin{enumerate} \item For any $A, B \in \alg$, $A \cap B \in \alg$. \item For any $A, B \in \alg$, $A \setminus B \in \alg$. - \end{enumerate} + \end\{enumerate\} + If $\alg$ is a $\sigma$-algebra, then: \begin{enumerate} \item[(1')] For any $\seq{A_n} \in \alg$, $\bigcap_{n \in \natp}A_n \in \alg$. diff --git a/src/measure/sets/borel.tex b/src/measure/sets/borel.tex index bdf5402..81d772a 100644 --- a/src/measure/sets/borel.tex +++ b/src/measure/sets/borel.tex @@ -98,7 +98,8 @@ \item Open sets of $X$. \item $\bracs{B(x, r)|x \in X, r > 0}$. \item $\bracsn{\ol{B(x, r)}|x \in X, r > 0}$. - \end{enumerate} + \end\{enumerate\} + \end{proposition} \begin{proof} diff --git a/src/measure/sets/elementary.tex b/src/measure/sets/elementary.tex index f03c006..3540dd8 100644 --- a/src/measure/sets/elementary.tex +++ b/src/measure/sets/elementary.tex @@ -8,7 +8,8 @@ \item[(P1)] $\emptyset \in \ce$. \item[(P2)] For any $A, B \in \ce$, $A \cap B \in \ce$. \item[(E)] For any $E, F \in \ce$ with $E \subset F$, there exists $\seqf{E_j} \subset \ce$ such that $E \setminus F = \bigsqcup_{j = 1}^n E_j$. - \end{enumerate} + \end\{enumerate\} + If $X \in \ce$, then (E) may be replaced with \begin{enumerate} \item[(E')] For any $E \in \ce$, there exists $\seqf{E_j} \subset \ce$ such that $E^c = \bigsqcup_{j = 1}^n E_j$. diff --git a/src/measure/sets/lambda.tex b/src/measure/sets/lambda.tex index 8967339..f546cf8 100644 --- a/src/measure/sets/lambda.tex +++ b/src/measure/sets/lambda.tex @@ -60,7 +60,8 @@ \braks{\bigcup_{n \in \natp}E_n} \cap F = \bigcup_{n \in \natp}E_n \cap F \in \lambda(\mathcal{P}) \] - \end{enumerate} + \end\{enumerate\} + so $\cm(\ce)$ is a $\lambda$-system. Since $\mathcal{P}$ is a $\pi$-system, $\cm(\mathcal{P}) \supset \mathcal{P}$, so $\cm(\mathcal{P}) = \lambda(\mathcal{P})$. Thus for any $E \in \lambda(\mathcal{P})$ and $F \in \lambda(\mathcal{P})$, $E \cap F \in \lambda(\mathcal{P})$. Therefore $\cm(\lambda(\mathcal{P})) \supset \mathcal{P}$, $\cm(\lambda(\mathcal{P})) = \lambda(\mathcal{P})$, and $\lambda(\mathcal{P})$ satisfies (P2). By \autoref{lemma:pi-lambda}, $\lambda(\mathcal{P})$ is a $\sigma$-algebra. diff --git a/src/measure/vector/complex.tex b/src/measure/vector/complex.tex index 3db5a49..de2404d 100644 --- a/src/measure/vector/complex.tex +++ b/src/measure/vector/complex.tex @@ -7,7 +7,8 @@ \begin{enumerate} \item[(M1)] $\mu(\emptyset) = 0$. \item[(M2)] For any $\seq{E_n} \subset \cm$ pairwise disjoint, $\mu\paren{\bigsqcup_{n \in \natp}E_n} = \sum_{n \in \natp} \mu(E_n)$ where the sum converges absolutely. - \end{enumerate} + \end\{enumerate\} + By \hyperref[Riemann's Rearrangement Theorem]{theorem:riemann-rearrangement}, (M2) implies that $\mu$ can only take at most one value in $\bracs{-\infty, \infty}$. \end{definition} @@ -101,7 +102,8 @@ \begin{enumerate} \item[(i)] For each $1 \le k \le n$, $M_k = m(A_k) - \mu(A_k) > 0$. \item[(ii)] For each $1 \le k \le n - 1$, $A_{k+1} \subset A_k$ with $\mu(A_{k+1}) - \mu(A_k) > M_{k}/2$. - \end{enumerate} + \end\{enumerate\} + Let $A_n \in \cm$ with $A_{n+1} \subset A_n$ such that $\mu(A_{n+1}) - \mu(A_n) > M_n/2$. Since $A_{n+1} \cap P = \emptyset$ and $\mu(P) = M$, $A_{n+1}$ cannot be positive, so \[ @@ -157,7 +159,8 @@ \item $\mu^+ \perp \mu^-$. \item $\mu = \mu^+ - \mu^-$. \item[(U)] For any other pair $(\nu^+, \nu^-)$ satisfying (1) and (2), $\mu^+ = \nu^+$ and $\mu^- = \nu^-$. - \end{enumerate} + \end\{enumerate\} + \end{theorem} \begin{proof} diff --git a/src/measure/vector/variation.tex b/src/measure/vector/variation.tex index de477d8..c5a4ce8 100644 --- a/src/measure/vector/variation.tex +++ b/src/measure/vector/variation.tex @@ -96,7 +96,8 @@ \begin{enumerate} \item $M(X, \cm; E)$ equipped with $\norm{\cdot}_{\text{var}}$ is a normed vector space over $K$. \item If $E$ is a Banach space, then so is $M(X, \cm; E)$. - \end{enumerate} + \end\{enumerate\} + \end{definition} \begin{proof} diff --git a/src/measure/vector/vector.tex b/src/measure/vector/vector.tex index 89cb6c5..e59f0c3 100644 --- a/src/measure/vector/vector.tex +++ b/src/measure/vector/vector.tex @@ -7,7 +7,8 @@ \begin{enumerate} \item[(M1)] $\mu(\emptyset) = 0$. \item[(M2)] For any $\seq{A_n} \subset \cm$ pairwise disjoint, $\mu\paren{\bigsqcup_{n \in \natp}A_n} = \sum_{n \in \natp} \mu(A_n)$ where the sum converges absolutely. - \end{enumerate} + \end\{enumerate\} + \end{definition} diff --git a/src/process/markov/definition.tex b/src/process/markov/definition.tex index e1b1e41..68eba24 100644 --- a/src/process/markov/definition.tex +++ b/src/process/markov/definition.tex @@ -31,7 +31,8 @@ \begin{enumerate} \item For each $t \ge 0$, $\bp_t \one = \one$. \item For each $t \ge 0$, $\bp_t$ is a positive linear functional. - \end{enumerate} + \end\{enumerate\} + known as the \textbf{semigroup} of $\bracs{P_t|t \ge 0}$. \end{definition} \begin{proof} diff --git a/src/topology/functions/set-systems.tex b/src/topology/functions/set-systems.tex index f325209..638e54d 100644 --- a/src/topology/functions/set-systems.tex +++ b/src/topology/functions/set-systems.tex @@ -36,7 +36,8 @@ \item $\mathfrak{E}(\sigma, \fU)$ generates a uniformity $\fV$ on $X^T$. \item The topology induced by $\fV$ is finer than the $\sigma$-open topology on $T^X$. \item If $\mathfrak{E}(\sigma, \fU)$ forms a fundamental system of entourages for $\fV$. - \end{enumerate} + \end\{enumerate\} + The uniformity $\fV$ is the \textbf{$\sigma$-uniformity}, and the topology induced by $\fV$ is the \textbf{topology of uniform convergence on the sets $\sigma$}/\textbf{$\sigma$-uniform topology} on $X^T$. \end{definition} \begin{proof} @@ -57,7 +58,8 @@ E(S, V) \circ E(S, V) \subset E(S, V \circ V) \subset E(S, U) \] - \end{enumerate} + \end\{enumerate\} + By \autoref{proposition:fundamental-entourage-criterion}, $\mathfrak{E}$ is a fundamental system of entourages for the uniformity that it generates. \end{proof} @@ -111,7 +113,8 @@ \item The product topology on $X^T$. \item The $\sigma$-open topology, where $\sigma$ is the collection of all finite sets. \item (If $X$ is a uniform space) The $\mathfrak{F}$-uniform topology, where $\fF = \bracs{F| F \subset X \text{ finite}}$. - \end{enumerate} + \end\{enumerate\} + This topology is the \textbf{topology of pointwise convergence} on $X^T$. \end{definition} \begin{proof} diff --git a/src/topology/functions/uniform.tex b/src/topology/functions/uniform.tex index 0dc7466..7da8788 100644 --- a/src/topology/functions/uniform.tex +++ b/src/topology/functions/uniform.tex @@ -23,7 +23,8 @@ \begin{enumerate} \item $C(X; Y) \subset Y^X$ is closed with respect to the uniform topology. \item If $X$ is a uniform space, then $UC(X; Y) \subset Y^X$ is closed with respect to the uniform topology. - \end{enumerate} + \end\{enumerate\} + In particular, if $Y$ is complete, then the above spaces are complete. \end{proposition} \begin{proof} @@ -49,3 +50,12 @@ If $Y$ is complete, then $Y^T$ with the uniform topology is complete by \autoref{proposition:set-uniform-complete}. Thus $C(T; X)$ and $UC(T; X)$ are both complete subspaces by \autoref{proposition:complete-closed}. \end{proof} + +\begin{corollary} +\label{corollary:uniform-limit-continuous-generated} + Let $X$ be a topological space, $\sigma \subset 2^X$ be an ideal such that $X$ is $\sigma$-generated, and $Y$ be a uniform space, then $C(X; Y) \subset Y^X$ is closed with respect to the $\sigma$-uniformity. +\end{corollary} +\begin{proof} + Let $f \in \overline{C(X; Y)} \subset Y^X$ with respect to the $\sigma$-uniformity. By \autoref{proposition:uniform-limit-continuous}, $f \in C(S; Y)$ for all $S \in \sigma$, so $f \in C(X; Y)$ by (3) of \autoref{definition:final-topology}. +\end{proof} + diff --git a/src/topology/main/baire.tex b/src/topology/main/baire.tex index 5d45d88..bcad652 100644 --- a/src/topology/main/baire.tex +++ b/src/topology/main/baire.tex @@ -9,7 +9,8 @@ \item For any $\seq{A_n} \subset 2^X$ nowhere dense, $\bigcup_{n \in \nat^+}A_n \subsetneq X$. \item For any $\seq{A_n} \subset 2^X$ closed with empty interior, $\bigcup_{n \in \nat^+}A_n$ has empty interior. \item For any $\seq{U_n} \subset 2^X$ open and dense, $\bigcap_{n \in \nat^+}U_n$ is dense. - \end{enumerate} + \end\{enumerate\} + If the above holds, then $X$ is a \textbf{Baire space}. \end{definition} \begin{proof} @@ -28,7 +29,8 @@ \begin{enumerate} \item[(a)] For all $n > 1$, $\ol V_n \subset U_n \cap V_{n - 1} \subset U$. \item[(b)] $\bigcap_{j \in \natp} \ol V_j$ is non-empty. - \end{enumerate} + \end\{enumerate\} + then $X$ is a Baire space. \end{lemma} \begin{proof} diff --git a/src/topology/main/compact.tex b/src/topology/main/compact.tex index fc66424..3bad35c 100644 --- a/src/topology/main/compact.tex +++ b/src/topology/main/compact.tex @@ -9,7 +9,8 @@ \item For every family $\seqi{E}$ of closed sets with $\bigcap_{j \in J}E_j \ne \emptyset$ for all $J \subset I$ finite, $\bigcap_{i \in I}E_i \ne \emptyset$. \item Every filter in $X$ has a cluster point. \item Every ultrafilter in $X$ converges. - \end{enumerate} + \end\{enumerate\} + If the above holds, then $X$ is \textbf{compact}. \end{definition} \begin{proof} @@ -89,7 +90,8 @@ \begin{enumerate} \item For any $x \in X$ and $U \in \cn_{X \times Y}^o(\bracs{x} \times Y)$, there exists $V \in \cn_X(x)$ such that $V \times Y \subset U$. \item For any $A \subset X$ and $U \in \cn_{X \times Y}^o(A \times Y)$, there exists $V \in \cn_X(A)$ such that $V \times Y \subset U$. - \end{enumerate} + \end\{enumerate\} + \end{lemma} \begin{proof} diff --git a/src/topology/main/connected.tex b/src/topology/main/connected.tex index 672f0a4..a0bf51b 100644 --- a/src/topology/main/connected.tex +++ b/src/topology/main/connected.tex @@ -8,7 +8,8 @@ \item For any $\emptyset \ne U, V \subset X$ open with $U \cup V = X$, $U \cap V \ne \emptyset$. \item There exists no surjective $f \in C(X; \bracs{0, 1})$. \item For any $U \subset X$ open and closed, either $U = \emptyset$ or $U = X$. - \end{enumerate} + \end\{enumerate\} + If the above holds, then $X$ is \textbf{connected}. \end{definition} \begin{proof} diff --git a/src/topology/main/continuity.tex b/src/topology/main/continuity.tex index e1d6791..8403807 100644 --- a/src/topology/main/continuity.tex +++ b/src/topology/main/continuity.tex @@ -8,7 +8,8 @@ \begin{enumerate} \item For each $V \in \cn(f(x))$, $f^{-1}(V) \in \cn(x)$. \item For each filter base $\fB \subset 2^X$ converging to $x$, $f(\fB)$ converges to $f(x)$. - \end{enumerate} + \end\{enumerate\} + If the above holds, then $f$ is \textbf{continuous at} $x \in X$. The following are also equivalent: @@ -16,7 +17,8 @@ \item For each $U \subset Y$ open, $f^{-1}(U)$ is open in $X$. \item $f$ is continuous at every $x \in X$. \item For each convergent filter base $\fB \subset 2^X$, $f(\fB)$ is convergent. - \end{enumerate} + \end\{enumerate\} + If the above holds, then $f$ is \textbf{continuous}. The collection $C(X; Y)$ is the space of all continuous functions from $X$ to $Y$. @@ -39,7 +41,8 @@ \begin{enumerate} \item[(a)] $\bigcup_{i \in I}U_i = X$. \item[(b)] For each $i, j \in I$, either $U_i \cap U_j = \emptyset$, or $f_i|_{U_i \cap U_j} = f_j|_{U_i \cap U_j}$. - \end{enumerate} + \end\{enumerate\} + then there exists a unique $f \in C(X; Y)$ such that $f|_{U_i} = f_i$ for all $i \in I$. \end{lemma} \begin{proof} diff --git a/src/topology/main/definition.tex b/src/topology/main/definition.tex index b86c2e3..13935dd 100644 --- a/src/topology/main/definition.tex +++ b/src/topology/main/definition.tex @@ -9,6 +9,7 @@ \item[(O2)] For any $U, V \in \topo$, $U \cap V \in \topo$. \item[(O3)] For any $\seqi{U} \subset \topo$, $\bigcup_{i \in I}U_i \in \topo$. \end{enumerate} + The elements of $\topo$ are known as \textbf{open sets}, and the pair $(X, \topo)$ is known as a \textbf{topological space}. \end{definition} @@ -47,6 +48,7 @@ \item For every $x \in X$, there exists $U \in \cb$ such that $x \in U$. \item For every $x \in X$ and $U \subset X$ open with $x \in U$, there exists $V \in \cb$ such that $x \in V \subset U$. \end{enumerate} + In which case, \[ \topo = \topo(\cb) = \bracs{\bigcup_{i \in I}U_i \bigg | \seqi{U} \subset \cb, I \text{ index set}} @@ -58,6 +60,7 @@ \item[(TB1)] For every $x \in X$, there exists $U \in \cb$ such that $x \in U$. \item[(TB2)] For every $x \in X$ and $U, V \in \cb$ such that $x \in U \cap V$, there exists $W \in \cb$ such that $x \in W \subset U \cap V$. \end{enumerate} + then $\topo(\cb)$ is a topology on $X$, and $\cb$ is a base for $\topo(\cb)$. \end{definition} \begin{proof} @@ -94,10 +97,10 @@ \begin{definition}[Initial Topology] \label{definition:initial-topology} - Let $X$ be a set, $\bracsn{(Y_j, \topo_i)}$ be a family of topological spaces, and $\seqi{f}$ be a family of maps such that $f_i: X \to Y_i$ for each $i \in I$, then there exists a topology $\topo$ on $X$ such that: + Let $X$ be a set, $\bracsn{(Y_i, \topo_i)}_{i \in I}$ be a family of topological spaces, and $\seqi{f}$ be a family of maps such that $f_i: X \to Y_i$ for each $i \in I$, then there exists a topology $\topo$ on $X$ such that: \begin{enumerate} \item For each $i \in I$, $f_i \in C(X; Y_i)$. - \item[(U)] If $\mathcal{S}$ is a topology on $X$ satisfying $(1)$, then $\mathcal{S} \supset \topo$. + \item[(U)] If $\mathcal{S}$ is a topology on $X$ satisfying (1), then $\mathcal{S} \supset \topo$. \item The family \[ @@ -107,13 +110,49 @@ is a base for $\topo$. \end{enumerate} - The topology $\topo$ is known a the \textbf{initial/weak topology} generated by the maps $\seqi{f}$. + + The topology $\topo$ is the \textbf{initial/weak topology} generated by the maps $\seqi{f}$. \end{definition} \begin{proof} - Let $\topo$ be the topology genereated by sets of the form $\ce = \bracs{f_i^{-1}(U_i)| i \in I, U_i \in \topo_i}$. Let $\topo$ be the topology generated by $\ce$, then + Let $\topo$ be the topology genereated by sets of the form $\ce = \bracs{f_i^{-1}(U_i)| i \in I, U_i \in \topo_i}$, then \begin{enumerate} \item For each $i \in I$, $\topo \supset \bracs{f_i^{-1}(U)|U \in \topo_i}$, so $f_i \in C(\topo; Y_i)$. \item If $\mathcal{S}$ is a topology such that $f_i \in C(X, \mathcal{S}; Y_i)$, then $\bracs{f_i^{-1}(U)|U \in \topo_i} \subset \mathcal{S}$. Thus $\ce \subset \mathcal{S}$ and $\mathcal{S} \supset \topo$. \item By \autoref{definition:generated-topology}, $\cb$ is a base for $\topo$. \end{enumerate} \end{proof} + +\begin{definition}[Final Topology] +\label{definition:final-topology} + Let $X$ be a set, $\bracsn{(Y_i, \topo_i)}_{i \in I}$ be a family of topological spaces, and $\seqi{f}$ be a family of maps such that $f_i: Y_i \to X$ for each $i \in I$, then there exists a topology $\topo$ on $X$ such that: + \begin{enumerate} + \item For each $i \in I$, $f_i \in C(Y_i; X)$. + \item[(U)] If $\mathcal{S}$ is a topology on $X$ satisfying (1), then $\mathcal{S} \subset \topo$. + \item For any topological space $Z$ and $F: X \to Z$, $F \in C(X; Z)$ if and only if $F \circ f_i \in C(Y_i; X)$ for all $i \in I$. + \end{enumerate} + + The topology $\topo$ is the \textbf{final topology} generated by the maps $\seqi{f}$. +\end{definition} +\begin{proof} + Let + \[ + \topo = \bracsn{U \subset X| f_i^{-1}(U) \in \topo_i \forall i \in I} + \] + + then since for each $i \in I$, $\topo_i$ is a topology on $Y_i$, $\topo$ is a topology on $X$. + + (1): By definition, for any $i \in I$ and $U \in \topo$, $f_i^{-1}(U) \in \topo_i$, so $f_i \in C(Y_i; X)$. + + (U): For any topology $\mathcal{S}$ satisfying (1) and $U \in \mathcal{S}$, $f_i^{-1}(U) \in \mathcal{T}_i$, so $\mathcal{S} \subset \mathcal{T}$. + + (3): Let $F: X \to Z$ such that $F \circ f_i \in C(Y_i; X)$ for all $i \in I$, then for any $U \subset Z$ open, $f_i^{-1}(F^{-1}(U)) \in \topo_i$ for all $i \in I$. Hence $F^{-1}(U) \in \topo$ and $F \in C(X; Z)$. +\end{proof} + +\begin{definition}[Generated Topology] +\label{definition:ideal-generated-topology} + Let $X$ be a topological space and $\sigma \subset 2^X$ be an ideal, then $X$ is \textbf{$\sigma$-generated} if the topology of $X$ is the final topology generated by $\bracs{\iota_S: S \to X|S \in \sigma}$. + + If $\kappa \subset 2^X$ is the collection of precompact sets of $X$, and $X$ is generated by $\kappa$, then $X$ is \textbf{compactly generated}. +\end{definition} + + diff --git a/src/topology/main/filters.tex b/src/topology/main/filters.tex index 4943f65..53bae52 100644 --- a/src/topology/main/filters.tex +++ b/src/topology/main/filters.tex @@ -25,7 +25,8 @@ \begin{enumerate} \item[(FB1)] For any $E, F \in \fB$, there exists $G \in \fB$ such that $G \subset E \cap F$. \item[(FB2)] $\emptyset \not\in \fB$. - \end{enumerate} + \end\{enumerate\} + Conversely, if $\fB \subset 2^X$ is a non-empty collection that satisfies (FB1) and (FB2), then $\fB$ is a base for the filter \[ \fF = \bracs{F \subset X| \exists E \in \fB: E \subset F} @@ -52,7 +53,8 @@ \begin{enumerate} \item $f(\fB) = \bracs{f(E)| E \in \fB}$ is also a filter base. \item If $\fB$ is an ultrafilter base, then $f(\fB)$ is also an ultrafilter base. - \end{enumerate} + \end\{enumerate\} + \end{proposition} \begin{proof} @@ -117,7 +119,8 @@ The smallest filter $\fF(\fB_0) \subset 2^X$ containing $\fB_0$ is the filter \t \item $\fF$ is maximal with respect to inclusion. \item For any $E \subset X$, either $E \in \fF$ or $E^c \in \fF$. \item For any $\seqf{F_j} \subset X$ such that $\bigcup_{j = 1}^n F_j \in \fF$, there exists $1 \le j \le n$ such that $F_j \in \fF$. - \end{enumerate} + \end\{enumerate\} + If the above holds, then $\fF$ is an \textbf{ultrafilter}. \end{definition} \begin{proof} @@ -157,7 +160,8 @@ The smallest filter $\fF(\fB_0) \subset 2^X$ containing $\fB_0$ is the filter \t \item $\fF \supset \cn(x)$. \item For each ultrafilter $\fU \supset \fF$, $\fU \supset \cn(x)$. \item There exists a fundamental system of neighbourhoods $\cb(x) \subset \cn(x)$ such that for every $E \in \cb$, there exists $F \in \fB$ with $F \subset E$. - \end{enumerate} + \end\{enumerate\} + If the above holds, then $x$ is a \textbf{limit point} of $\fB$, and $\fB$ \textbf{converges} to $x$. If $A \subset X$ and $\fB \subset 2^A$, then $\fB$ \textbf{converges} to $x$ if $\fF(\fB) \supset \bracsn{U \cap A| U \in \cn(x)}$. @@ -178,7 +182,8 @@ The smallest filter $\fF(\fB_0) \subset 2^X$ containing $\fB_0$ is the filter \t \item $x \in \bigcap_{E \in \fF}\overline{E}$. \item There exists a fundamental system of neighbourhoods $\cb(x) \subset \cn(x)$ such that for every $E \in \cb$ and $f \in \fB$, $E \cap F \ne \emptyset$. \item There exists a filter $\fU \supset \fB$ that converges to $x$. - \end{enumerate} + \end\{enumerate\} + If the above holds, then $x$ is a \textbf{cluster/accumulation point} of $\fB$. In particular, if $\fF$ is an ultra filter, then (6) implies that the limit points and cluster points of $\fF$ coincide. \end{definition} \begin{proof} diff --git a/src/topology/main/hausdorff.tex b/src/topology/main/hausdorff.tex index 623b569..cc11a69 100644 --- a/src/topology/main/hausdorff.tex +++ b/src/topology/main/hausdorff.tex @@ -12,7 +12,8 @@ \item Every filter in $X$ converges to at most one point. \item For any index set $I$, the diagonal $\Delta$ is closed in $X^I$. \item The diagonal $\Delta$ is closed in $X \times X$. - \end{enumerate} + \end\{enumerate\} + If the above holds, then $X$ is a \textbf{T2/Hausdorff} space. \end{definition} \begin{proof} diff --git a/src/topology/main/lch.tex b/src/topology/main/lch.tex index 71da91a..7e90ccd 100644 --- a/src/topology/main/lch.tex +++ b/src/topology/main/lch.tex @@ -8,7 +8,8 @@ \item For any $x \in X$, there exists $K \in \cn(x)$ compact. \item For any $x \in X$, $\cn(x)$ admits a fundamental system of neighbourhoods consisting of compact sets. \item For any $x \in X$, $\cn(x)$ admits a fundamental system of neighbourhoods consisting of precompact sets. - \end{enumerate} + \end\{enumerate\} + If the above holds, then $X$ is a \textbf{locally compact Hausdorff (LCH)} space. \end{definition} \begin{proof} @@ -76,6 +77,35 @@ then by the \hyperref[gluing lemma for continuous functions]{lemma:gluing-continuous}, $\ol F \in C_c(X; \real)$ with $\ol F|_K = F|_K = f$ and $\supp{F} \subset \supp{\eta} \subset V \subset U$. \end{proof} +\begin{proposition} +\label{proposition:lch-compactly-generated} + Let $X$ be a LCH space, then: + \begin{enumerate} + \item $X$ is compactly generated. + \item For any uniform space $Y$, $C(X; Y) \subset Y^X$ is closed with respect to the compact-open topology. + \end{enumerate} +\end{proposition} +\begin{proof} + (1): Let $U \subset X$ such that $U \cap K$ is open in $K$ for all $K \subset X$ compact. For any $x \in U$, there exists a compact neighbourhood $K \in \cn(x)$. In which case, $U \supset U \cap K \in \cn(x)$, so $U \in \cn(x)$ for all $x \in U$. By \autoref{lemma:openneighbourhood}, $U$ is open. + + (2): By \autoref{proposition:compact-uniform-open}, the compact-open topology coincides with the compact-uniform topology on $C(X; Y)$. Since $X$ is compactly generated, $C(X; Y) \subset Y^X$ is closed with respect to the compact-open topology by \autoref{corollary:uniform-limit-continuous-generated}. +\end{proof} + + +\begin{proposition} +\label{proposition:lch-product} + Let $\seqi{X}$ be a family of LCH spaces. If $X_i$ is compact for all but finitely many $i \in I$, then $X = \prod_{i \in I}X_i$ is a LCH space. +\end{proposition} +\begin{proof} + By \autoref{proposition:product-hausdorff}, $\prod_{i \in I}X_i$ is Hausdorff. Let $x \in \prod_{i \in I}X_i$ and $i \in I$. If $X_i$ is not compact, let $U_i \in \cn_{X_i}(\pi_i(x))$ be compact. Otherwise, let $U_i = X_i$. Let $U = \prod_{i \in I}U_i$, then since $U_i \ne X_i$ for only finitely many $i \in I$, $U \in \cn_X(x)$. By \hyperref[Tychonoff's Theorem]{theorem:tychonoff}, $U$ is compact. Therefore $X$ is locally compact. +\end{proof} + + +\subsection{Paracompactness and LCH Spaces} +\label{subsection:lch-paracompact} + + + \begin{proposition}[{{\cite[Proposition 4.39]{Folland}}}] \label{proposition:lch-sigma-compact} Let $X$ be a LCH space, then the following are equivalent: @@ -92,7 +122,8 @@ \item[(a)] For each $0 \le k \le n$, $U_k$ is a precompact open set. \item[(b)] For each $0 \le k < n$, $\overline{U_k} \subset U_{k+1}$. \item[(c)] For each $1 \le k \le n$, $U_k \supset \bigcup_{j = 1}^k K_j$. - \end{enumerate} + \end\{enumerate\} + By \autoref{lemma:lch-compact-neighbour}, there exists $U_{n+1} \in \cn^o(\overline{U_n} \cup K_{n+1})$ precompact. In which case, by (c), \[ U_{n+1} \supset \ol{U_n} \cup K_{n+1} \supset \bigcup_{j = 1}^n K_j \cup K_{n+1} = \bigcup_{j = 1}^{n+1}K_j @@ -154,7 +185,8 @@ \begin{enumerate} \item[(a)] $\ol{E} \subset \bigcup_{x \in X_E}N_x$. \item[(b)] For every $x \in X_E$, $N_x \cap E \ne \emptyset$. - \end{enumerate} + \end\{enumerate\} + Let $X_{\ce} = \bigcup_{E \in \ce}X_\ce$, and for each $E \in \ce$, let \[ G_E = \bigcup_{\substack{x \in X_\ce \\ \ol{N_x} \subset E}}N_x @@ -238,12 +270,4 @@ Let $x \in X$, then there exists $n \in \natp$ such that $x \in U_n \setminus U_{n-1}$. In which case, if $n > 1$, then $x \in U_{n} \setminus \ol{U_{n - 2}} = V_{n-1}$. If $n = 1$, then $x \in U_{2} = V_1$. Thus $\seq{V_n}$ is an open cover of $X$. In addition, for any $m, n \in \natp$ with $m \le n$, $V_m \cap V_n \ne \emptyset$ implies that $n - m < 2$, so $\seq{V_n}$ is locally finite. By (2) of \autoref{proposition:lch-paracompact}, $X$ is paracompact. \end{proof} -\begin{proposition} -\label{proposition:lch-product} - Let $\seqi{X}$ be a family of LCH spaces. If $X_i$ is compact for all but finitely many $i \in I$, then $X = \prod_{i \in I}X_i$ is a LCH space. -\end{proposition} -\begin{proof} - By \autoref{proposition:product-hausdorff}, $\prod_{i \in I}X_i$ is Hausdorff. Let $x \in \prod_{i \in I}X_i$ and $i \in I$. If $X_i$ is not compact, let $U_i \in \cn_{X_i}(\pi_i(x))$ be compact. Otherwise, let $U_i = X_i$. Let $U = \prod_{i \in I}U_i$, then since $U_i \ne X_i$ for only finitely many $i \in I$, $U \in \cn_X(x)$. By \hyperref[Tychonoff's Theorem]{theorem:tychonoff}, $U$ is compact. Therefore $X$ is locally compact. -\end{proof} - diff --git a/src/topology/main/neighbourhoods.tex b/src/topology/main/neighbourhoods.tex index 901b4ed..2df02af 100644 --- a/src/topology/main/neighbourhoods.tex +++ b/src/topology/main/neighbourhoods.tex @@ -36,12 +36,14 @@ \item[(F2)] For any $A, B \in \cn_\topo(x)$, $A \cap B \in \cn_\topo(x)$. \item[(V1)] For every $A \in \cn_\topo(x)$, $x \in A$. \item[(V2)] For every $V \in \cn_\topo(x)$, there exists $W \in \cn_\topo(x)$ such that $V \in \cn_\topo(y)$ for all $y \in W$. - \end{enumerate} + \end\{enumerate\} + Conversely, if $\cn: X \to 2^X$ is a mapping such that \begin{enumerate} \item $\cn(x) \ne \emptyset$ for all $x \in X$. \item $\cn(x)$ satisfies (F1), (F2), (V1), and (V2). - \end{enumerate} + \end\{enumerate\} + then there exists a unique topology $\topo \subset 2^X$ such that $\cn = \cn_\topo$. \end{proposition} diff --git a/src/topology/main/normal.tex b/src/topology/main/normal.tex index 725e193..e2574df 100644 --- a/src/topology/main/normal.tex +++ b/src/topology/main/normal.tex @@ -23,7 +23,8 @@ \begin{enumerate} \item[(a)] $U_1 = B^c$. \item[(b)] For any $p, q \in \rational \cap [0, 1]$ with $p < q$, $\overline{U_p} \subset U_q$. - \end{enumerate} + \end\{enumerate\} + \item There exists $f \in C(X; [0, 1])$ with $f|_A = 0$ and $f|_B = 1$. \end{enumerate} \end{lemma} diff --git a/src/topology/main/product.tex b/src/topology/main/product.tex index 9473137..73f2e0e 100644 --- a/src/topology/main/product.tex +++ b/src/topology/main/product.tex @@ -40,7 +40,7 @@ \begin{proposition} \label{proposition:productfilterconvergence} -Let $\bracsn{(X_i, \topo_i)}_{i \in I}$ be a family of topological spaces and $\B$ be a filter base on $\prod_{i \in I}X_i$, then $\fB$ converges to $x \in \prod_{i \in I}X_i$ if and only if $\pi_i(\fB)$ converges to $\pi_i(x)$ for all $i \in I$. +Let $\bracsn{(X_i, \topo_i)}_{i \in I}$ be a family of topological spaces and $\fB$ be a filter base on $\prod_{i \in I}X_i$, then $\fB$ converges to $x \in \prod_{i \in I}X_i$ if and only if $\pi_i(\fB)$ converges to $\pi_i(x)$ for all $i \in I$. \end{proposition} \begin{proof} $(\Rightarrow)$: Let $i \in I$ and $U \in \cn(\pi_i(x))$, then $\pi_i^{-1}(U) \in \cn(x)$. Since $\fB$ converges to $x$, there exists $B \in \fB$ with $B \subset \pi_i^{-1}(U)$. In which case, $\pi_i(B) \subset U$ and $\pi_i(\fB)$ converges to $\pi_i(x)$. diff --git a/src/topology/main/quotient.tex b/src/topology/main/quotient.tex index 941a710..a24c2a7 100644 --- a/src/topology/main/quotient.tex +++ b/src/topology/main/quotient.tex @@ -12,7 +12,8 @@ \begin{enumerate} \item For any $U \subset Y$, $U$ is open if and only if $\pi^{-1}(U)$ is open. \item $\pi \in C(X; Y)$, and for any $U \subset X$ saturated and open, $\pi(U)$ is open. - \end{enumerate} + \end\{enumerate\} + If the above holds, then $\pi$ is a \textbf{quotient map}. \end{definition} \begin{proof} @@ -37,7 +38,8 @@ \] \item $\pi$ is a quotient map. - \end{enumerate} + \end\{enumerate\} + The space $(\td X, \pi)$ is the \textbf{quotient} of $X$ by $\sim$. \end{definition} \begin{proof} diff --git a/src/topology/main/regular.tex b/src/topology/main/regular.tex index 6ecdad9..1061745 100644 --- a/src/topology/main/regular.tex +++ b/src/topology/main/regular.tex @@ -7,7 +7,8 @@ \begin{enumerate} \item For each $x \in X$ and $A \subset X$ closed with $x \not\in A$, there exists $U \in \cn(x)$ and $V \in \cn(A)$ such that $U \cap V = \emptyset$. \item For each $x \in X$, the closed neighbourhoods of $x$ forms a fundamental system of neighbourhoods at $x$. - \end{enumerate} + \end\{enumerate\} + If $X$ is a T1 space such that the above holds, then $X$ is \textbf{regular}. \end{definition} \begin{proof}[Proof {{\cite[Proposition 1.4.11]{Bourbaki}}}. ] diff --git a/src/topology/metric/metric.tex b/src/topology/metric/metric.tex index 17fce1a..c80a0ea 100644 --- a/src/topology/metric/metric.tex +++ b/src/topology/metric/metric.tex @@ -9,7 +9,8 @@ \item[(M)] For any $x, y \in X$ with $x \ne y$, $d(x, y) > 0$. \item[(PM2)] For any $x, y \in X$, $d(x, y) = d(y, x)$. \item[(PM3)] For any $x, y, z \in X$, $d(x, z) \le d(x, y) + d(y, z)$. - \end{enumerate} + \end\{enumerate\} + The pair $(X, d)$ is a \textbf{metric space}, which comes with the metric uniformity induced by $d$, and the corresponding topology. \end{definition} diff --git a/src/topology/uniform/cauchy.tex b/src/topology/uniform/cauchy.tex index 31bea99..4e3ed07 100644 --- a/src/topology/uniform/cauchy.tex +++ b/src/topology/uniform/cauchy.tex @@ -12,7 +12,8 @@ \begin{enumerate} \item $A, B$ are $V$-small. \item $A \cap B \ne \emptyset$. - \end{enumerate} + \end\{enumerate\} + then $A \cup B$ is $V \circ V$-small. \end{lemma} \begin{proof} diff --git a/src/topology/uniform/completion.tex b/src/topology/uniform/completion.tex index fdee20c..ce27f61 100644 --- a/src/topology/uniform/completion.tex +++ b/src/topology/uniform/completion.tex @@ -19,7 +19,8 @@ Moreover, if $f \in UC(X; Y)$, then $F \in UC(\wh X; Y)$. - \end{enumerate} + \end\{enumerate\} + Moreover, \begin{enumerate} \item[(4)] For any symmetric entourage $V \in \fU$, let @@ -46,14 +47,16 @@ \item[(FB1)] Let $\wh U, \wh V \in \wh \fB$. By \autoref{lemma:symmetricfundamentalentourage}, there exists a symmetric entourage $W \in \fU$ such that $W \subset U \cap V$. In which case, for any $(\fF, \mathfrak{G}) \in \wh W$, there exists $E \in \fF \cap \mathfrak{G}$ with $E \times E \subset W \subset U \cap V$. Thus $\wh W \subset \wh U \cap \wh V$. \item[(UB1)] Let $\wh U \in \wh \fB$ and $\fF \in \wh X$, then since $\fF$ is Cauchy, there exists $E \in \fF$ such that $E \times E \subset U$, so $(\fF, \fF) \in \wh U$. \item[(UB2)] Let $\wh U \in \wh \fB$. By \autoref{lemma:symmetricfundamentalentourage}, there exists a symmetric entourage $W \in \fU$ such that $W \circ W \subset U$. For any $\fF, \mathfrak{G}, \mathfrak{H} \in \wh X$ such that $(\fF, \mathfrak{G}), (\mathfrak{G}, \mathfrak{H}) \in \wh W$, there exists $W$-small sets $E \in \fF \cap \mathfrak{G}$ and $F \in \fF \cap \mathfrak{H}$. Since $\mathfrak{G}$ is a filter, $E \cap F \ne \emptyset$ by (F2) and (F3). By \autoref{lemma:small-intersect}, $E \cup H$ is $W \circ W$-small and thus $U$-small. Using (F1), $E \cup H \in \fF \cap \mathfrak{H}$, so $(\fF, \mathfrak{H}) \in \wh U$. Therefore $\wh W \circ \wh W \subset U$. - \end{enumerate} + \end\{enumerate\} + By \autoref{proposition:fundamental-entourage-criterion}, there exists a unique uniformity $\wh \fU \supset \wh \fB$. Moreover, $\wh \fB$ consists of symmetric entourages by construction. (1, Hausdorff): It is sufficient to show that $\Delta$ is closed and use (6) of \autoref{definition:hausdorff}. Let $(\fF, \mathfrak{G}) \in \ol{\Delta}$, then $(\fF, \mathfrak{G}) \in U$ for all $U \in \fU$ closed. Let $\fB = \bracs{F \cup G| F \in \fF, G \in \mathfrak{G}}$, then \begin{enumerate} \item[(FB1)] For any $F \cup G, F' \cup G' \in \fB$, $(F \cup G) \cap (F' \cup G') \supset (F \cap F') \cup (G \cap G') \in \fB$. \item[(FB2)] By (F3), $\emptyset \not\in \fF \cup \mathfrak{G}$, so $\emptyset \not\in \fB$. - \end{enumerate} + \end\{enumerate\} + Thus $\fB$ is a filter base by \autoref{proposition:filterbasecriterion}, and the filter $\mathfrak{H}$ generated by $\fB$ is contained in $\fF$ and $\mathfrak{G}$. By \autoref{proposition:goodentourages}, for every $U \in \fU$, there exists a $U$-small set $E \in \fF \cap \mathfrak{G} \subset \fB \subset \mathfrak{H}$. So $\mathfrak{H} \subset \fF, \mathfrak{G}$ is a Cauchy filter. By minimality of $\fF$ and $\mathfrak{G}$, $\fF = \mathfrak{G} = \mathfrak{H}$. (2): For each $x \in X$, $\cn(x)$ is a minimal Cauchy filter by (1) of \autoref{proposition:cauchyfilterlimit}. Define $\iota: X \to \wh X$ by $x \mapsto \cn(x)$. Let $\wh U \in \wh \fU$ and $(\cn(x), \cn(y)) \in \wh U$, then there exists a $U$-small set $E \in \cn(x) \cap \cn(y)$. By (V1), $(x, y) \in E \times E \in U$. @@ -104,7 +107,8 @@ } \] - \end{enumerate} + \end\{enumerate\} + known as the \textbf{Hausdorff uniform space associated with} $(X, \fU)$. \end{definition} \begin{proof}[Proof {{\cite[Proposition 2.8.16]{Bourbaki}}}. ] @@ -167,7 +171,8 @@ Moreover, $\ol{F}(\wh X) = \overline{F(X)}$, and $\ol{F}$ is an embedding. - \end{enumerate} + \end\{enumerate\} + In particular, by \autoref{proposition:dense-product}, there is a natural isomorphism \[ \prod_{i \in I}\wh X_i \iso \wh{\prod_{i \in I}X_i} diff --git a/src/topology/uniform/definition.tex b/src/topology/uniform/definition.tex index 9735594..934b3a2 100644 --- a/src/topology/uniform/definition.tex +++ b/src/topology/uniform/definition.tex @@ -41,7 +41,8 @@ \item[(U1)] For every $U \in \fU$, $U \supset \Delta = \bracs{(x, x)| x \in X}$. \item[(U2)] For any $U \in \fU$, $U^{-1} \in \fU$. \item[(U3)] For any $U \in \fU$, there exists $V \in \fU$ such that $V \circ V \subset U$. - \end{enumerate} + \end\{enumerate\} + The elements of $\fU$ are called the \textbf{entourages} of $\fU$, and the pair $(X, \fU)$ is a \textbf{uniform space}. For any $x, y \in X$ and $U \in \fU$, $x$ and $y$ are \textbf{$U$-close} if $(x, y) \in U$. @@ -79,7 +80,8 @@ \item[(FB1)] For each $U, V \in \fB$, there exists $W \in \fB$ such that $W \subset U \cap V$. \item[(UB1)] For each $V \in \fB$, $\Delta \subset V$. \item[(UB2)] For each $V \in \fB$, there exists $W \in \fB$ such that $W \circ W \subset V$. - \end{enumerate} + \end\{enumerate\} + then there exists a unique uniformity $\fU \subset 2^{X \times X}$, which is given by \[ \fU = \bracs{U \subset X \times X| \exists V \in \fB: V \subset U} @@ -176,7 +178,8 @@ V = (\bracs{x} \times V)(x) = (U \cup (\bracs{x} \times V))(x) = W(x) \in \cn(x) \begin{enumerate} \item $V \circ M \circ V \in \cn(M)$. \item Let $\fB$ be the set of all symmetric entourages, then $\ol{M} = \bigcap_{V \in \fB}V \circ M \circ V$. - \end{enumerate} + \end\{enumerate\} + with respect to the product topology on $X \times X$. \end{proposition} \begin{proof} @@ -223,7 +226,8 @@ V = (\bracs{x} \times V)(x) = (U \cup (\bracs{x} \times V))(x) = W(x) \in \cn(x) \begin{enumerate} \item $\mathfrak{O} = \bracs{U^o| U \in \fU}$ \item $\mathfrak{K} = \bracsn{\overline{U}| U \in \fU}$. - \end{enumerate} + \end\{enumerate\} + By \autoref{lemma:symmetricfundamentalentourage}, there exists fundamental systems of entourages for $\fU$ consisting of symmetric and open/closed sets. \end{proposition} \begin{proof} @@ -267,7 +271,8 @@ V = (\bracs{x} \times V)(x) = (U \cup (\bracs{x} \times V))(x) = W(x) \in \cn(x) \item $X$ is Hausdorff. \item $X$ is regular. \item $\Delta = \bigcap_{U \in \fU}U$. - \end{enumerate} + \end\{enumerate\} + If the above holds, then $X$ is \textbf{separated}. \end{definition} \begin{proof} diff --git a/src/topology/uniform/equicontinuous.tex b/src/topology/uniform/equicontinuous.tex index 509f43a..da54876 100644 --- a/src/topology/uniform/equicontinuous.tex +++ b/src/topology/uniform/equicontinuous.tex @@ -66,4 +66,19 @@ \end{proof} +\begin{corollary} +\label{corollary:arzela-locally-compact} + Let $X$ be a LCH space, $Y$ be a uniform space, and $\cf \subset C(X; Y)$ such that: + \begin{enumerate}[label=(E\arabic*)] + \item $\cf$ is equicontinuous. + \item For each $x \in X$, $\cf(x) = \bracs{f(x)|f \in \cf}$ is precompact in $Y$. + \end{enumerate} + + then $\cf$ is a precompact subset of $C(X; Y)$ with respect to the compact uniformity. +\end{corollary} +\begin{proof} + By the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli}, $\cf$ is a precompact subset of $Y^X$. By \autoref{proposition:lch-compactly-generated}, $C(X; Y)$ is a closed subset of $Y^X$. Therefore $\cf$ is a precompact subset of $C(X; Y)$. +\end{proof} + + diff --git a/src/topology/uniform/metric.tex b/src/topology/uniform/metric.tex index 7fd9fa0..fec8aeb 100644 --- a/src/topology/uniform/metric.tex +++ b/src/topology/uniform/metric.tex @@ -10,11 +10,13 @@ The axioms of uniform spaces strongly resembles working in a metric space. In fa \item[(PM1)] For any $x \in X$, $d(x, x) = 0$. \item[(PM2)] For any $x, y \in X$, $d(x, y) = d(y, x)$. \item[(PM3)] For any $x, y, z \in X$, $d(x, z) \le d(x, y) + d(y, z)$. - \end{enumerate} + \end\{enumerate\} + If $d$ satisfies the above and \begin{enumerate} \item[(M)] For any $x, y \in X$ with $x \ne y$, $d(x, y) > 0$. - \end{enumerate} + \end\{enumerate\} + then $d$ is a \textbf{metric}. \end{definition} @@ -62,7 +64,8 @@ The axioms of uniform spaces strongly resembles working in a metric space. In fa \item For each $U \subset X$, $U$ is open if and only if for every $x \in U$, there exists $J \subset I$ finite and $r > 0$ such that $\bigcap_{j \in J}B_j(x, r) \subset U$. \item For each $i \in I$, $d_i \in UC(X \times X; [0, \infty))$. \item[(U)] For any other uniformity $\mathfrak{V}$ satisfying (4), $\mathfrak{U} \subset \mathfrak{V}$. - \end{enumerate} + \end\{enumerate\} + The uniformity $\fU$ is the \textbf{pseudometric uniformity} induced by $\seqi{d}$, and the topology induced by $\fU$ is the \textbf{pseudometric topology} on $X$ induced by $\seqi{d}$. \end{definition} \begin{proof} @@ -112,7 +115,8 @@ The axioms of uniform spaces strongly resembles working in a metric space. In fa \item[(a)] $U_{0} = X \times X$. \item[(b)] For each $n \in \natz$, $U_n$ is symmetric. \item[(c)] For each $n \in \natz$, $U_{n + 1} \circ U_{n+1} \subset U_n$. - \end{enumerate} + \end\{enumerate\} + then there exists a pseudometric $d: X \times X \to [0, 1]$ such that \[ U_{n+1} \subset E(d, 2^{-n}) \subset U_{n-1} @@ -141,7 +145,8 @@ The axioms of uniform spaces strongly resembles working in a metric space. In fa \] As this holds for all such $\seqf{x_j}$ and $\seqf[m]{y_j}$, $d(x, z) \le d(x, y) + d(y, z)$. - \end{enumerate} + \end\{enumerate\} + so $d$ is a pseudometric. For any $(x, y) \in U_{n+1}$, $d(x, y) \le \rho(x, y) < 2^{-n}$, so $U_{n+1} \subset E(d, 2^{-n})$. @@ -250,7 +255,8 @@ The axioms of uniform spaces strongly resembles working in a metric space. In fa \item[(a)] For each $1 \le k \le n$, $V_k$ is symmetric. \item[(b)] For each $1 \le k \le n$, $V_k \subset U_k$. \item[(c)] For each $1 \le k < n$, $V_{k+1} \circ V_{k+1} \subset V_{k}$. - \end{enumerate} + \end\{enumerate\} + Let $W = V_n \cap U_{n+1}$, then by \autoref{lemma:symmetricfundamentalentourage}, there exists $V_{n+1} \in \fU$ symmetric such that $V_{n+1} \circ V_{n+1} \subset W$. Thus $\bracs{V_k|1 \le k \le n + 1} \subset \fU$ satisfies (a), (b), and (c) for $n + 1$. Let $V_0 = X \times X$, then by \autoref{lemma:uniform-sequence-pseudometric}, there exists a pseudometric $d: X \times X \to [0, \infty)$ such that for each $n \in \natp$, diff --git a/src/topology/uniform/uc.tex b/src/topology/uniform/uc.tex index aa8f036..c09757b 100644 --- a/src/topology/uniform/uc.tex +++ b/src/topology/uniform/uc.tex @@ -7,7 +7,8 @@ \begin{enumerate} \item For every $V \in \mathfrak{V}$, there exists $V' \in \fU$ such that $(f(x), f(y)) \in V$ whenever $(x, y) \in V'$. \item For every $V \in \mathfrak{V}$, $(f \times f)^{-1}(V) \in \fU$. - \end{enumerate} + \end\{enumerate\} + If the above holds, then $f$ is a \textbf{uniformly continuous} function. The collection $UC(X; Y)$ denotes the set of all uniformly continuous functions from $X$ to $Y$. @@ -34,7 +35,8 @@ \begin{enumerate} \item For each $i \in I$, $f_i \in UC(X; Y_i)$. \item[(U)] If $\mathfrak{V}$ is a uniformity on $X$ satisfying $(1)$, then $\mathfrak{V} \supset \fU$. - \end{enumerate} + \end\{enumerate\} + Moreover, \begin{enumerate} \item[(3)] The family