diff --git a/.vscode/project.code-snippets b/.vscode/project.code-snippets index fa25880..acf01f3 100644 --- a/.vscode/project.code-snippets +++ b/.vscode/project.code-snippets @@ -27,6 +27,17 @@ "$0" ] }, + "Summary": { + "scope": "latex", + "prefix": "summ", + "body": [ + "\\begin{summary}[$1]", + "\\label{summary:$2}", + " $3", + "\\end{summary}", + "$0" + ] + }, "Lemma Block": { "scope": "latex", "prefix": "lem", diff --git a/preamble.sty b/preamble.sty index a9281ac..731529c 100644 --- a/preamble.sty +++ b/preamble.sty @@ -15,6 +15,7 @@ \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} +\newtheorem{summary}[theorem]{Summary} % \newtheorem{exercise}[subsection]{Exercise} % \newtheorem{situation}[subsection]{Situation} diff --git a/src/cat/cat/index.tex b/src/cat/cat/index.tex index 38f5bd1..17c0924 100644 --- a/src/cat/cat/index.tex +++ b/src/cat/cat/index.tex @@ -5,4 +5,4 @@ \input{./cat-func.tex} \input{./universal.tex} -\input{./tensor.tex} +\input{./tensor.tex} \ No newline at end of file diff --git a/src/cat/cat/tensor.tex b/src/cat/cat/tensor.tex index 8887d97..7d8757e 100644 --- a/src/cat/cat/tensor.tex +++ b/src/cat/cat/tensor.tex @@ -1,8 +1,172 @@ -\section{The Tensor Product} -\label{section:tensor-product} +\section{Universal Constructions for Modules} +\label{section:universal-module} +\begin{definition}[Product] +\label{definition:product-module} + Let $R$ be a ring and $\seqi{A}$ be $R$-modules, then there exists $(A, \bracsn{\pi_i}_{i \in I})$ such that: + \begin{enumerate} + \item For each $i \in I$, $\pi_i \in \hom(A; A_i)$. + \item[(U)] For any $(B, \seqi{T})$ satisfying (1), there exists a unique $T \in \hom(B; A)$ such that the following diagram commutes: + \[ + \xymatrix{ + B \ar@{->}[rd]_{T_i} \ar@{->}[r]^{T} & A \ar@{->}[d]^{\pi_i} \\ + & A_i + } + \] + \end{enumerate} + + The module $A = \prod_{i \in I}A_i$ is the \textbf{product} of $\seqi{A}$. +\end{definition} + +\begin{definition}[Direct Sum] +\label{definition:direct-sum} + Let $E$ be a ring and $\seqi{A}$ be $R$-modules, then there exists $(A, \bracsn{\iota_i}_{i \in I})$ such that: + \begin{enumerate} + \item For each $i \in I$, $\iota_i \in \hom(A_i; A)$. + \item[(U)] For each $(B, \seqi{T})$ satisfying (1), there exists a unique $T \in \hom(A; B)$ such that the following diagram commutes + \[ + \xymatrix{ + A \ar@{->}[r]^{T} & B \\ + A_i \ar@{->}[u]^{\iota_i} \ar@{->}[ru]_{T_i} & + } + \] + \end{enumerate} + + The module $A = \bigoplus_{i \in I}A_i$ is the \textbf{direct sum} of $\seqi{A}$. +\end{definition} +\begin{proof} + Let + \[ + A = \bracs{x \in \prod_{i \in I}A_i \bigg | x_i \ne 0 \quad \text{for finitely many}\ i \in I} + \] + + For each $i \in I$, let + \[ + \iota_i: A_i \to A \quad (\iota_ix)_j = \begin{cases} + x &i = j \\ + 0 &i \ne j + \end{cases} + \] + + then $\iota_i \in \hom(A_i; A)$. + + (U): Let + \[ + T: A \to B \quad x \mapsto \sum_{i \in I}T_ix_i + \] + + then $T \in \hom(A; B)$ and the diagram commutes. Since $\bigcup_{i \in I}\iota_i(A_i)$ spans $A$, $T$ is the unique linear map making the diagram commute. +\end{proof} + +\begin{proposition} +\label{proposition:module-direct-limit} + Let $R$ be a ring and $(\seqi{A}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system of $R$-modules, then there exists $(A, \bracsn{T^i_A}_{i \in I})$ such that: + \begin{enumerate} + \item For each $i \in I$, $T^i_A \in \hom({A_i; A})$. + \item For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes: + + \[ + \xymatrix{ + A_i \ar@{->}[rd]_{T^i_A} \ar@{->}[r]^{T^i_j} & A_j \ar@{->}[d]^{T^j_A} \\ + & A + } + \] + + + \item[(U)] For any pair $(B, \bracsn{S^i_B}_{i \in I})$ satisfying (1) and (2), there exists a unique $S \in \hom({A, B})$ such that the following diagram commutes + + \[ + \xymatrix{ + A_i \ar@{->}[d]_{T^i_A} \ar@{->}[rd]^{S^i_B} & \\ + A \ar@{->}[r]_{g} & B + } + \] + + + for all $i \in I$. + \end{enumerate} +\end{proposition} +\begin{proof} + Let $M = \bigoplus_{i \in I}A_i$. For any $i, j \in I$ with $i \lesssim j$ and $x \in A_i$, let $x_{i, j} \in M$ such that for any $k \in I$, + \[ + \pi_k(x_{i, j}) = \begin{cases} + x &k = i \\ + T^i_j x &k = j \\ + 0 &k \ne i, j + \end{cases} + \] + + Let $N \subset M$ be the submodule generated by $\bracs{x_{i, j}|i, j \in I, i \lesssim j, x \in A_i}$, $A = M/N$, and $\pi: M \to M/N$ be the canonical map. + + (1): For each $i \in I$, let + \[ + T^i_M: A_i \to M \quad \pi_k T^i_M x = \begin{cases} + x &k = i \\ + 0 &k \ne i + \end{cases} + \] + + and $T^i_A = \pi \circ T^i_M$. + + (2): Let $i, j \in I$ with $i \lesssim j$, then for any $x \in A_i$, $T^i_Mx - T^j_M T^i_j x \in N$. Hence $T^i_Ax = T^j_A T^i_jx$. + + (U): Let + \[ + S_0: M \to B \quad x \mapsto \sum_{i \in I}S^i_B \pi_i x + \] + + then $S_0$ is the unique linear map such that $S_0 \circ T^i_M = S^i_B$ for all $i \in I$. For any $i, j \in I$ with $i \lesssim J$, $S^i_B x = S^j_B T^i_j x$, so $\ker S_0 \supset N$. By the first isomorphism theorem, there exists a unique $S \in \hom(A; B)$ such that $S_0 = S \circ \pi$. +\end{proof} + +\begin{proposition} +\label{proposition:module-inverse-limit} + Let $R$ be a ring and $(\seqi{A}, \bracs{T^i_j|i, j \in I, i \lesssim j})$ be a downward-directed system of $R$-modules, then there exists $(A, \bracsn{T^A_i}_{i \in I})$ such that: + \begin{enumerate} + \item For each $i \in I$, $T^A_i \in \hom(A; A_i)$. + \item For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes: + \[ + \xymatrix{ + A_i \ar@{->}[r]^{T^i_j} & A_j \\ + A \ar@{->}[u]^{T^A_i} \ar@{->}[ru]_{T^A_j} & + } + \] + + \item[(U)] For any pair $(B, \bracsn{S^B_i}_{i \in I})$ satisfying (1) and (2), there exists a unique $S \in \hom(B; A)$ such that the following diagram commutes + + \[ + \xymatrix{ + & A_i \\ + B \ar@{->}[r]_{S} \ar@{->}[ru]^{S^B_i} & A \ar@{->}[u]_{T^A_i} + } + \] + + + for all $i \in I$. + \end{enumerate} +\end{proposition} +\begin{proof} + Let + \[ + A = \bracs{x \in \prod_{i \in I}A_i \bigg | \pi_j(x) = T^i_j\pi_i(x) \forall i, j \in I, i \lesssim j} + \] + + For each $i \in I$, let $T^A_i = \pi_i$, then $(A, \bracsn{T^A_i}_{i \in I})$ satisfies (1) and (2) by definition of $A$. + + (U): Let $(B, \bracsn{S^B_i}_{i \in I})$ satisfying (1) and (2). Let + \[ + S: B \to \prod_{i \in I}A_i \quad \pi_i(Sx) = S^B_i + \] + + then for any $x \in B$ and $i, j \in I$ with $i \lesssim j$, + \[ + \pi_j (Sx) = S^B_jx = T^i_j S^B_ix = T^i_j \pi_i(S x) + \] + + so $S \in \hom(B; A)$, and the diagram commutes. Since any map $f: B \to A$ is uniquely determined by its composition with the projections, $S$ is unique. +\end{proof} + \begin{definition}[Tensor Product] \label{definition:tensor-product} Let $R$ be a commutative ring and $\seqf{E_j}$ be $R$ modules, then there exists a pair $(\bigotimes_{j = 1}^n E_j, \iota)$ such that: diff --git a/src/cat/cat/universal.tex b/src/cat/cat/universal.tex index 8d66f10..784ba85 100644 --- a/src/cat/cat/universal.tex +++ b/src/cat/cat/universal.tex @@ -147,112 +147,5 @@ Let $\catc$ be a category and $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim \end{enumerate} \end{definition} -\begin{proposition} -\label{proposition:module-direct-limit} - Let $R$ be a ring and $(\seqi{A}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system of $R$-modules, then there exists $(A, \bracsn{T^i_A}_{i \in I})$ such that: - \begin{enumerate} - \item For each $i \in I$, $T^i_A \in \hom({A_i, A})$. - \item For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes: - - \[ - \xymatrix{ - A_i \ar@{->}[rd]_{T^i_A} \ar@{->}[r]^{T^i_j} & A_j \ar@{->}[d]^{T^j_A} \\ - & A - } - \] - - - \item[(U)] For any pair $(B, \bracsn{S^i_B}_{i \in I})$ satisfying (1) and (2), there exists a unique $S \in \hom({A, B})$ such that the following diagram commutes - - \[ - \xymatrix{ - A_i \ar@{->}[d]_{T^i_A} \ar@{->}[rd]^{S^i_B} & \\ - A \ar@{->}[r]_{g} & B - } - \] - - - for all $i \in I$. - \end{enumerate} -\end{proposition} -\begin{proof} - Let $M = \bigoplus_{i \in I}A_i$. For any $i, j \in I$ with $i \lesssim j$ and $x \in A_i$, let $x_{i, j} \in M$ such that for any $k \in I$, - \[ - \pi_k(x_{i, j}) = \begin{cases} - x &k = i \\ - T^i_j x &k = j \\ - 0 &k \ne i, j - \end{cases} - \] - - Let $N \subset M$ be the submodule generated by $\bracs{x_{i, j}|i, j \in I, i \lesssim j, x \in A_i}$, $A = M/N$, and $\pi: M \to M/N$ be the canonical map. - - (1): For each $i \in I$, let - \[ - T^i_M: A_i \to M \quad \pi_k T^i_M x = \begin{cases} - x &k = i \\ - 0 &k \ne i - \end{cases} - \] - - and $T^i_A = \pi \circ T^i_M$. - - (2): Let $i, j \in I$ with $i \lesssim j$, then for any $x \in A_i$, $T^i_Mx - T^j_M T^i_j x \in N$. Hence $T^i_Ax = T^j_A T^i_jx$. - - (U): Let - \[ - S_0: M \to B \quad x \mapsto \sum_{i \in I}S^i_B \pi_i x - \] - - then $S_0$ is the unique linear map such that $S_0 \circ T^i_M = S^i_B$ for all $i \in I$. For any $i, j \in I$ with $i \lesssim J$, $S^i_B x = S^j_B T^i_j x$, so $\ker S_0 \supset N$. By the first isomorphism theorem, there exists a unique $S \in \hom(A; B)$ such that $S_0 = S \circ \pi$. -\end{proof} - -\begin{proposition} -\label{proposition:module-inverse-limit} - Let $R$ be a ring and $(\seqi{A}, \bracs{T^i_j|i, j \in I, i \lesssim j})$ be a downward-directed system of $R$-modules, then there exists $(A, \bracsn{T^A_i}_{i \in I})$ such that: - \begin{enumerate} - \item For each $i \in I$, $T^A_i \in \hom(A; A_i)$. - \item For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes: - \[ - \xymatrix{ - A_i \ar@{->}[r]^{T^i_j} & A_j \\ - A \ar@{->}[u]^{T^A_i} \ar@{->}[ru]_{T^A_j} & - } - \] - - \item[(U)] For any pair $(B, \bracsn{S^B_i}_{i \in I})$ satisfying (1) and (2), there exists a unique $S \in \hom(B; A)$ such that the following diagram commutes - - \[ - \xymatrix{ - & A_i \\ - B \ar@{->}[r]_{S} \ar@{->}[ru]^{S^B_i} & A \ar@{->}[u]_{T^A_i} - } - \] - - - for all $i \in I$. - \end{enumerate} -\end{proposition} -\begin{proof} - Let - \[ - A = \bracs{x \in \prod_{i \in I}A_i \bigg | \pi_j(x) = T^i_j\pi_i(x) \forall i, j \in I, i \lesssim j} - \] - - For each $i \in I$, let $T^A_i = \pi_i$, then $(A, \bracsn{T^A_i}_{i \in I})$ satisfies (1) and (2) by definition of $A$. - - (U): Let $(B, \bracsn{S^B_i}_{i \in I})$ satisfying (1) and (2). Let - \[ - S: B \to \prod_{i \in I}A_i \quad \pi_i(Sx) = S^B_i - \] - - then for any $x \in B$ and $i, j \in I$ with $i \lesssim j$, - \[ - \pi_j (Sx) = S^B_jx = T^i_j S^B_ix = T^i_j \pi_i(S x) - \] - - so $S \in \hom(B; A)$, and the diagram commutes. Since any map $f: B \to A$ is uniquely determined by its composition with the projections, $S$ is unique. -\end{proof} - diff --git a/src/fa/lc/barrel.tex b/src/fa/lc/barrel.tex new file mode 100644 index 0000000..7dbe5c4 --- /dev/null +++ b/src/fa/lc/barrel.tex @@ -0,0 +1,67 @@ +\section{Barreled Spaces} +\label{section:barrel} + +\begin{definition}[Barrel] +\label{definition:barrel} + Let $E$ be a TVS over $K \in \RC$ and $D \subset E$, then $D$ is a \textbf{barrel} if it is convex, circled, radial, and closed. +\end{definition} + +\begin{definition}[Barreled Space] +\label{definition:barreled-space} + Let $E$ be a locally convex space over $K \in \RC$, then the following are equivalent: + \begin{enumerate} + \item The barrels of $E$ forms a fundamental system of neighbourhoods at $0$. + \item Every barrel in $E$ is a neighbourhood of $0$. + \item Every lower semicontinuous seminorm on $E$ is continuous. + \end{enumerate} +\end{definition} +\begin{proof} + $(2) \Rightarrow (1)$: Let $\fB \subset \cn_E(0)$ be a fundamental system of neighbourhoods at $0$ consisting of convex, circled, and radial sets, then $\ol{\fB} = \bracsn{\ol U|U \in \fB}$ is a fundamental system of neighbourhoods at $0$ consisting of barrels. + + $(2) \Rightarrow (3)$: Let $\rho: E \to [0, \infty)$ be a lower semicontinuous seminorm, then $\bracs{\rho > 1}$ is open and $\bracs{\rho \le 1}$ is a Barrel. In which case, $\rho$ is continuous by (4) of \autoref{lemma:continuous-seminorm}. + + $(3) \Rightarrow (2)$: Let $D \subset E$ be a barrel and $\rho: E \to [0, \infty)$ be its gauge. By (4) of \autoref{definition:gauge}, $D = \bracs{\rho \le 1}$, so $\bracs{\rho > 1}$ is open, and $\rho$ is semicontinuous. By assumption, $\rho$ is continuous, so $D \in \cn_E(0)$ by (5) of \autoref{lemma:continuous-seminorm}. +\end{proof} + +\begin{summary}[Barreled Spaces] +\label{summary:barreled-space} + The following types of locally convex spaces are barreled: + \begin{enumerate} + \item Every locally convex space with the Baire property. + \item Every Banach space and every Fréchet space. + \item Inductive limits of barreled spaces. + \item Spaces of type (LB) and (LF). + \item The locally convex direct sum of barreled spaces. + \item Products of barreled spaces. + \end{enumerate} +\end{summary} +\begin{proof} + (1), (2): \autoref{proposition:baire-barrel}. + + (3), (4), (5): \autoref{proposition:barrel-limit}. + + (6): TODO. +\end{proof} + + +\begin{proposition} +\label{proposition:baire-barrel} + Let $E$ be a locally convex space over $K \in \RC$. If $E$ is a Baire space, then $E$ is barreled. +\end{proposition} +\begin{proof}[Proof, {{\cite[II.7.1]{SchaeferWolff}}}. ] + Let $D \subset E$ be a Barrel, then $E = \bigcup_{n \in \natp}nD$ is a countable union of closed sets. Since $E$ is Baire, there exists $n \in \natp$, $U \in \cn_E(0)$ circled, and $x \in E$ such that $x + U \in nB$. In which case, + \[ + U \subset (x + U) - (x + U) \subset nB - nB = 2nB + \] + + so $2nB$ and thus $B$ is a neighbourhood of $0$. +\end{proof} + +\begin{proposition} +\label{proposition:barrel-limit} + Let $\seqi{E}$ be locally convex spaces over $K \in \RC$, $E$ be a vector space over $K$, and $\seqi{T}$ such that $T_i \in \hom(E_i; E)$ for all $i \in I$, then the inductive locally convex topology on $E$ induced by $\seqi{T}$ is barreled. +\end{proposition} +\begin{proof}[Proof, {{\cite[II.7.2]{SchaeferWolff}}}] + Let $D \subset E$ be a barrel, then for each $i \in I$, $T_i^{-1}(D) \subset E_i$ is also a barrel, and thus a neighbourhood of $0$ in $E_i$. By (5) of \autoref{definition:lc-inductive}, $D$ is a neighbourhood of $0$ in $E$, so $E$ is barreled. +\end{proof} + diff --git a/src/fa/lc/convex.tex b/src/fa/lc/convex.tex index e54a207..15a7af3 100644 --- a/src/fa/lc/convex.tex +++ b/src/fa/lc/convex.tex @@ -118,15 +118,16 @@ \item $[\cdot]$ is continuous. \item $[\cdot]$ is continuous at $0$. \item $\bracs{x \in E| [x] < 1} \in \cn_E(0)$. + \item $\bracs{x \in E| [x] \le 1} \in \cn_E(0)$. \end{enumerate} \end{lemma} \begin{proof} - $(4) \Rightarrow (1)$: Let $x, y \in E$ and $r > 0$. If + $(5) \Rightarrow (1)$: Let $x, y \in E$ and $r > 0$. If \[ - x - y \in \bracs{x \in E|[x] < r} = r\bracs{x \in E|[x] < 1} \in \cn_E(0) + x - y \in \bracs{x \in E|[x] \le r} = r\bracs{x \in E|[x] \le 1} \in \cn_E(0) \] - then $[x - y] < r$. + then $[x - y] \le r$. \end{proof} @@ -147,7 +148,7 @@ \begin{definition}[Gauge/Minkowski Functional] \label{definition:gauge} - Let $E$ be a vector space over $K \in \RC$ and $A \subset E$ be a radial set, then the mapping + Let $E$ be a vector space over $K \in \RC$ and $A \subset E$ be radial, then the mapping \[ [\cdot]_A: E \to [0, \infty) \quad x \mapsto \inf\bracsn{\lambda > 0| \lambda^{-1}x \in A} \] @@ -157,11 +158,12 @@ \item For any $x \in E$ and $\lambda \ge 0$, $[\lambda x]_A = \lambda [x]_A$. \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} In particular, - \begin{enumerate} - \item[(4)] If $A$ is convex, then $[\cdot]_A$ is a sublinear functional. - \item[(5)] If $A$ is convex and circled, then $[\cdot]_A$ is a seminorm. + \begin{enumerate}[start=4] + \item If $A$ is convex, then $[\cdot]_A$ is a sublinear functional. + \item If $A$ is convex and circled, then $[\cdot]_A$ is a seminorm. \end{enumerate} \end{definition} \begin{proof} @@ -171,6 +173,13 @@ \] then $(\lambda + \mu)^{-1} \in A$, and $\lambda + \mu \ge [x + y]_A$. Thus $[x + y]_A \le [x]_A + [y]_A$. + + (4): Let $x \in \bracs{\rho \le 1}$, then $\lambda x \in A$ for all $\lambda \in (0, 1)$. Therefore + \[ + x \in \overline{\bracs{\lambda x|\lambda \in (0, 1)}} \subset A + \] + + so $x \in \overline{A}$. \end{proof} \begin{definition}[Locally Convex Space] diff --git a/src/fa/lc/hahn-banach.tex b/src/fa/lc/hahn-banach.tex index 067860b..fa998ad 100644 --- a/src/fa/lc/hahn-banach.tex +++ b/src/fa/lc/hahn-banach.tex @@ -140,7 +140,7 @@ \end{enumerate} \end{proposition} \begin{proof} - (1): Let $\rho_M: E \to [0, \infty)$ be the quotient of $\rho$ by $M$, then $\rho_M \le \rho$ is a continuous seminorm on $E$ by \autoref{definition:quotient-norm}. Let $\phi_0: Kx \to K$ be defined by $\lambda x \mapsto \lambda \rho_M(x)$. By the \hyperref[Hahn-Banach Theorem]{theorem:hahn-banach}, there exists $\phi \in \hom{E; K}$ such that $\dpb{x, \phi}{E} = \rho_M(x)$ and $|\phi| \le \rho_M \le \rho$. + (1): Let $\rho_M: E \to [0, \infty)$ be the quotient of $\rho$ by $M$, then $\rho_M \le \rho$ is a continuous seminorm on $E$ by \autoref{definition:quotient-norm}. Let $\phi_0: Kx \to K$ be defined by $\lambda x \mapsto \lambda \rho_M(x)$. By the \hyperref[Hahn-Banach Theorem]{theorem:hahn-banach}, there exists $\phi \in \hom(E; K)$ such that $\dpb{x, \phi}{E} = \rho_M(x)$ and $|\phi| \le \rho_M \le \rho$. (2): By (1) applied to $M = \bracs{0}$. diff --git a/src/fa/lc/index.tex b/src/fa/lc/index.tex index 77ece48..1a65c95 100644 --- a/src/fa/lc/index.tex +++ b/src/fa/lc/index.tex @@ -4,6 +4,7 @@ \input{./convex.tex} \input{./continuous.tex} +\input{./barrel.tex} \input{./bornologic.tex} \input{./quotient.tex} \input{./projective.tex} diff --git a/src/fa/lc/inductive.tex b/src/fa/lc/inductive.tex index 1643234..cee34ed 100644 --- a/src/fa/lc/inductive.tex +++ b/src/fa/lc/inductive.tex @@ -3,7 +3,7 @@ \begin{definition}[Inductive Locally Convex Topology] \label{definition:lc-inductive} - Let $\seqi{E}$ be locally convex spaces over $K \in \RC$, $\seqi{T}$ such that $T_i \in \hom(E_i; E)$ for all $i \in I$, and $E$ be a vector space over $K$, then there exists a topology $\topo$ on $E$ such that: + Let $\seqi{E}$ be locally convex spaces over $K \in \RC$, $E$ be a vector space over $K$, and $\seqi{T}$ such that $T_i \in \hom(E_i; E)$ for all $i \in I$, then there exists a topology $\topo$ on $E$ such that: \begin{enumerate} \item $(E, \topo)$ is a locally convex space over $K$. \item For each $i \in I$, $T_i \in L(E_i; E)$. @@ -11,10 +11,16 @@ \item For any locally convex space $F$ 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 $i \in I$. \item The family \[ - \mathcal{B} = \bracs{U \subset E|U \text{ convex, radial, circled}, T_i^{-1}(U) \in \cn_{E_i}(0) \forall i \in I} + \mathcal{B} = \bracs{U \subset E|U \text{ convex, circled, radial}, T_i^{-1}(U) \in \cn_{E_i}(0) \forall i \in I} \] is a fundamental system of neighbourhoods for $E$ at $0$. + \item If $E$ is spanned by $\bigcup_{i \in I}T_i(E_i)$, then + \[ + \fB = \bracs{\Gamma\paren{\bigcup_{i \in I}T_i(U_i)} \bigg | U_i \in \cn_{E_i}(0)} + \] + + is a fundamental system of neighbourhoods for $E$ at $0$. \end{enumerate} The topology $\topo$ is the \textbf{inductive locally convex topology} on $E$ induced by $\seqi{T}$. \end{definition} @@ -31,7 +37,43 @@ (U): Let $U \in \cn_{(E, \mathcal{S})}(0)$ be convex, circled, and radial. By (2), $T_i^{-1}(U) \in \cn_{E_i}(0)$ for all $i \in I$. Thus the convex, circled, and radial neighbourhoods of $0$ in $(E, \mathcal{S})$ is a subset of $\mathcal{B}$. - (4): Let $i \in I$ and $U \in \cn_F(0)$ be convex, circled, and radial. Since $T \circ E_i \in L(E_i; F)$, $T_i^{-1}(T^{-1}(U)) \in \cn_{E_i}(0)$, so $T^{-1}(U) \in \mathcal{B} \subset \cn_E(0)$. + (5): Let $i \in I$ and $U \in \cn_F(0)$ be convex, circled, and radial. Since $T \circ E_i \in L(E_i; F)$, $T_i^{-1}(T^{-1}(U)) \in \cn_{E_i}(0)$, so $T^{-1}(U) \in \mathcal{B} \subset \cn_E(0)$. + + (6): If $E$ is spanned by $\bigcup_{i \in I}T_i(E_i)$, then each set in $\fB$ is radial. Hence $\fB$ is a family of neighbourhoods of $E$ at $0$. + + Let $U \in \cn_E(0)$ be convex, circled, and radial, then for each $i \in I$, $T_i^{-1}(U) \in \cn_{E_i}(0)$, so $U \supset \bigcup_{i \in I}T_i[T_i^{-1}(U)]$. Since $U$ is convex and circled, $U \supset \Gamma\paren{\bigcup_{i \in I}T_i[T_i^{-1}(U)]} \in \fB$. Therefore $\fB$ forms a fundamental system of neighbourhoods for $E$ at $0$. +\end{proof} + +\begin{definition}[Locally Convex Direct Sum] +\label{definition:lc-direct-sum} + Let $\seqi{E}$ be locally convex spaces over $K \in \RC$, then there exists $(E, \seqi{\iota})$ such that: + \begin{enumerate} + \item $E$ is a locally convex space over $K$. + \item For each $i \in I$, $\iota_i \in L(E_i; E)$. + \item[(U)] For each $(F, \seqi{T})$ satisfying (1) and (2), there exists a unique $T \in L(E; F)$ such that the following diagram commutes: + \[ + \xymatrix{ + A \ar@{->}[r]^{T} & B \\ + A_i \ar@{->}[u]^{\iota_i} \ar@{->}[ru]_{T_i} & + } + \] + + \item The family + \[ + \fB = \bracs{\Gamma\paren{\bigcup_{i \in I}\iota_i(U_i)} \bigg | U_i \in \cn_{E_i}(0)} + \] + + is a fundamental system of neighbourhoods for $E$ at $0$. + \end{enumerate} + + The space $E = \bigoplus_{i \in I}E_i$ is the \textbf{locally convex direct sum} of $\seqi{E}$. +\end{definition} +\begin{proof} + Let $(E, \seqi{\iota})$ be the direct sum of $\seqi{E}$ as vector spaces, and equip it with the inductive topology induced by $\seqi{\iota}$, then $(E, \seqi{\iota})$ satisfies (1) and (2). + + (U): By (U) of the \hyperref[direct sum]{definition:direct-sum}, there exists a unique $T \in \hom(E; F)$ such that the diagram commutes. In which case, by (4) of \autoref{definition:lc-inductive}, $T \in L(E; F)$. + + (4): By (6) of \autoref{definition:lc-inductive}. \end{proof} \begin{definition}[Inductive Limit] @@ -88,6 +130,9 @@ In addition to the neighbourhood construction given above, the inductive topology may also be constructed as the weak topology generated by all topologies satisfying certain properties. While this more non-constructive method is simpler, it does not directly provide an explicit fundamental system of neighbourhoods at $0$. \end{remark} +\subsection{Strict Inductive Limits} +\label{subsection:lc-induct-strict} + \begin{lemma}[{{\cite[Lemma II.6.4]{SchaeferWolff}}}] \label{lemma:lc-induct-separate} Let $E$ be a locally convex space over $K$, $M \subset E$ be a subspace, $U \in \cn_M(0)$ be convex and circled, then diff --git a/src/fa/rs/bv.tex b/src/fa/rs/bv.tex index ecbbaf3..34b00c1 100644 --- a/src/fa/rs/bv.tex +++ b/src/fa/rs/bv.tex @@ -66,7 +66,7 @@ \item $BV([a, b]; E)$ is a vector space. \item For each continuous seminorm $\rho$ on $E$, $[\cdot]_{\text{var}, \rho}$ is a seminorm on $BV([a, b]; E)$. \item Let $\fF$ be a filter on $BV([a, b]; E)$ and $f: [a, b] \to E$. If - \begin{enumerate} + \begin{enumerate}[label=\alph*] \item $\pi_x(\fF) \to f(x)$ for all $x \in [a, b]$. \item For every continuous seminorm $\rho$ on $E$, there exists $U \in \fF$ such that $\sup_{g \in U}[g]_{\text{var}, \rho} = M_\rho < \infty$. \end{enumerate}