From 16e6beb1174819f56a31c7d2b0a8ff6d8c108a68 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Tue, 17 Mar 2026 15:18:31 -0400 Subject: [PATCH] Replaced mentions of normed spaces to normed vector spaces. --- src/fa/lp/definition.tex | 10 +++++----- src/fa/norm/absolute.tex | 6 +++--- src/fa/norm/index.tex | 2 +- src/fa/norm/multilinear.tex | 2 +- src/fa/norm/normed.tex | 6 +++--- src/fa/norm/separable.tex | 6 +++--- src/fa/rs/bv.tex | 4 ++-- src/measure/measurable-maps/metric.tex | 2 +- src/measure/radon/c0.tex | 2 +- src/measure/radon/radon.tex | 2 +- src/measure/vector/variation.tex | 2 +- src/measure/vector/vector.tex | 4 ++-- 12 files changed, 24 insertions(+), 24 deletions(-) diff --git a/src/fa/lp/definition.tex b/src/fa/lp/definition.tex index 4678824..546ac47 100644 --- a/src/fa/lp/definition.tex +++ b/src/fa/lp/definition.tex @@ -63,7 +63,7 @@ \begin{theorem}[Minkowski's Inequality, {{\cite[6.5]{Folland}}}] \label{theorem:minkowski} - Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed space, $p \in [1, \infty]$, and $f, g \in \mathcal{L}^p(X; E)$, then + Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed vector space, $p \in [1, \infty]$, and $f, g \in \mathcal{L}^p(X; E)$, then \[ \norm{f + g}_{L^p(X; E)} \le \norm{f}_{L^p(X; E)} + \norm{g}_{L^p(X; E)} \] @@ -89,7 +89,7 @@ \begin{definition}[$L^p$ Space] \label{definition:lp} - Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed space, and $p \in [1, \infty]$, then $\norm{\cdot}_{L^p(X; E)}$ is a seminorm on $\mathcal{L}^p(X; E)$. The quotient + Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed vector space, and $p \in [1, \infty]$, then $\norm{\cdot}_{L^p(X; E)}$ is a seminorm on $\mathcal{L}^p(X; E)$. The quotient \[ L^p(X, \cm, \mu; E) = \mathcal{L}^p(X, \cm, \mu; E)/\bracs{f|f = 0\text{ a.e.}} \] @@ -121,7 +121,7 @@ \begin{proposition} \label{proposition:lp-simple-dense} - Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed space, and $p \in [1, \infty)$, then $\Sigma(X, \cm; E) \cap L^p(X; E)$ is dense in $L^p(X; E)$. + Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed vector space, and $p \in [1, \infty)$, then $\Sigma(X, \cm; E) \cap L^p(X; E)$ is dense in $L^p(X; E)$. \end{proposition} \begin{proof} Let $f \in L^p(X; E)$. By \autoref{definition:strongly-measurable}, there exists $\seq{f_n} \subset \Sigma(X, \cm; E)$ such that $\norm{f_n}_E \le \norm{f}_E$ and $\norm{f_n - f}_E \to 0$ strongly pointwise as $n \to \infty$. By the \hyperref[Dominated Convergence Theorem]{proposition:dct-lp}, $f_n \to f$ in $L^p(X; E)$. @@ -129,7 +129,7 @@ \begin{theorem}[Markov's Inequality] \label{theorem:markov-inequality} - Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed space, and $f: X \to E$ be a Borel measurable function, then + Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed vector space, and $f: X \to E$ be a Borel measurable function, then \begin{enumerate} \item For any $\alpha > 0$, \[ @@ -158,7 +158,7 @@ \begin{proposition} \label{proposition:lp-in-measure} - Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed space, $p \in [1, \infty]$, $\seq{f_n} \subset L^p(X; E)$, and $f \in L^p(X; E)$ such that $f_n \to f$ in $L^p$, then $f_n \to f$ in measure. + Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed vector space, $p \in [1, \infty]$, $\seq{f_n} \subset L^p(X; E)$, and $f \in L^p(X; E)$ such that $f_n \to f$ in $L^p$, then $f_n \to f$ in measure. \end{proposition} \begin{proof} Let $\eps > 0$. If $p < \infty$, then by \hyperref[Markov's Inequality]{theorem:markov-inequality}, diff --git a/src/fa/norm/absolute.tex b/src/fa/norm/absolute.tex index 0fbbf37..cfb4638 100644 --- a/src/fa/norm/absolute.tex +++ b/src/fa/norm/absolute.tex @@ -3,12 +3,12 @@ \begin{definition}[Absolute Convergence] \label{definition:absolute-convergence} - Let $E$ be a normed space, then a series $\sum_{n = 1}^\infty x_n$ with $\seq{x_n} \subset E$ \textbf{converges absolutely} if $\sum_{n \in \natp}\norm{x_n}_E < \infty$. + Let $E$ be a normed vector space, then a series $\sum_{n = 1}^\infty x_n$ with $\seq{x_n} \subset E$ \textbf{converges absolutely} if $\sum_{n \in \natp}\norm{x_n}_E < \infty$. \end{definition} \begin{lemma} \label{lemma:banach-absolute} - Let $E$ be a normed space, then the following are equivalent: + Let $E$ be a normed vector space, then the following are equivalent: \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 $x = \sum_{n = 1}^\infty x_n$. @@ -24,7 +24,7 @@ \begin{definition}[Unconditional Convergence] \label{definition:unconditional-convergence} - Let $E$ be a normed space, then a series $\sum_{n = 1}^\infty x_n$ with $\seq{x_n} \subset E$ \textbf{converges unconditionally} if for any bijection $\sigma: \natp \to \natp$, + Let $E$ be a normed vector space, then a series $\sum_{n = 1}^\infty x_n$ with $\seq{x_n} \subset E$ \textbf{converges unconditionally} if for any bijection $\sigma: \natp \to \natp$, \[ \sum_{n = 1}^\infty x_n = \sum_{n = 1}^\infty x_{\sigma(n)} \] diff --git a/src/fa/norm/index.tex b/src/fa/norm/index.tex index 0480ae7..b0c0f16 100644 --- a/src/fa/norm/index.tex +++ b/src/fa/norm/index.tex @@ -1,4 +1,4 @@ -\chapter{Normed Spaces} +\chapter{Normed Vector Spaces} \label{chap:normed-spaces} \input{./normed.tex} diff --git a/src/fa/norm/multilinear.tex b/src/fa/norm/multilinear.tex index b7cec3d..f2ad6e8 100644 --- a/src/fa/norm/multilinear.tex +++ b/src/fa/norm/multilinear.tex @@ -3,7 +3,7 @@ \begin{proposition} \label{proposition:bilinear-separate} - Let $E, F, G$ be normed spaces and $T: E \times F \to G$ be a bilinear map. If: + Let $E, F, G$ be normed vector spaces and $T: E \times F \to G$ be a bilinear map. If: \begin{enumerate} \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 $G$. diff --git a/src/fa/norm/normed.tex b/src/fa/norm/normed.tex index b8231cd..bbffa96 100644 --- a/src/fa/norm/normed.tex +++ b/src/fa/norm/normed.tex @@ -34,7 +34,7 @@ \begin{theorem}[Successive Approximation] \label{theorem:successive-approximation} - Let $E, F$ be normed spaces, $T \in L(E; F)$, $C \ge 0$, and $\gamma \in (0, 1)$. If for all $y \in F$, there exists $x \in E$ such that: + Let $E, F$ be normed vector spaces, $T \in L(E; F)$, $C \ge 0$, and $\gamma \in (0, 1)$. If for all $y \in F$, there exists $x \in E$ such that: \begin{enumerate} \item[(a)] $\norm{x}_E \le C\norm{y}_F$. \item[(b)] $\norm{y - Tx}_F \le \gamma \norm{y}_F$. @@ -69,7 +69,7 @@ \begin{theorem}[Uniform Boundedness Principle] \label{theorem:uniform-boundedness} - Let $E, F$ be normed spaces and $\mathcal{T} \subset L(E; F)$. If + Let $E, F$ be normed vector spaces and $\mathcal{T} \subset L(E; F)$. If \begin{enumerate} \item For every $x \in E$, $\sup_{T \in \mathcal{T}}\norm{Tx}_F < \infty$. \item $E$ is a Banach space. @@ -90,7 +90,7 @@ \begin{proposition} \label{proposition:dual-norm} - Let $E$ be a normed space, then for any $x \in E$, + Let $E$ be a normed vector space, then for any $x \in E$, \[ \norm{x}_E = \sup_{\substack{\phi \in E^* \\ \norm{\phi}_{E^*} = 1}}\dpn{x, \phi}{E} \] diff --git a/src/fa/norm/separable.tex b/src/fa/norm/separable.tex index 0e44b4e..ebc9bf4 100644 --- a/src/fa/norm/separable.tex +++ b/src/fa/norm/separable.tex @@ -1,9 +1,9 @@ -\section{Separable Normed Spaces} +\section{Separable Normed Vector Spaces} \label{section:separable-banach-space} \begin{proposition} \label{proposition:separable-dual} - Let $E$ be a separable normed space, then $E^*$ is separable with respect to the weak*-topology. + Let $E$ be a separable normed vector space, then $E^*$ is separable with respect to the weak*-topology. \end{proposition} \begin{proof} Let $\seq{x_n} \subset E$ be a dense subset and $S = \bracsn{\phi \in E^*| \norm{\phi}_{E^*} \le 1}$. For each $N \in \natp$, let @@ -23,7 +23,7 @@ \begin{proposition} \label{proposition:separable-banach-borel-sigma-algebra} - Let $E$ be a separable normed space, then the Borel $\sigma$-algebra on $E$ is generated by the following families of sets: + Let $E$ be a separable normed vector space, then the Borel $\sigma$-algebra on $E$ is generated by the following families of sets: \begin{enumerate} \item Open sets in $E$ with respect to the strong topology. \item $\bracs{B(x, r)|x \in E, r > 0}$. diff --git a/src/fa/rs/bv.tex b/src/fa/rs/bv.tex index 6d12ec7..74fbb99 100644 --- a/src/fa/rs/bv.tex +++ b/src/fa/rs/bv.tex @@ -16,7 +16,7 @@ is the \textbf{total variation} of $f$ on $[a, b]$ with respect to $\rho$. - If $E$ is a normed space, then the variation and total variation of $f$ is taken with respect to its norm. + If $E$ is a normed vector space, then the variation and total variation of $f$ is taken with respect to its norm. \end{definition} \begin{definition}[Bounded Variation, {{\cite[Proposition X.1.1]{Lang}}}] @@ -35,7 +35,7 @@ then $f \in BV([a, b]; E)$ with $[f]_{\text{var}, \rho} \le M_\rho$. \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} - If $(E, \norm{\cdot}_E)$ is a normed space, then + If $(E, \norm{\cdot}_E)$ is a normed vector space, then \begin{enumerate} \item[(5)] $f$ has at most countably many discontinuities. \end{enumerate} diff --git a/src/measure/measurable-maps/metric.tex b/src/measure/measurable-maps/metric.tex index 776c4c1..c39ad91 100644 --- a/src/measure/measurable-maps/metric.tex +++ b/src/measure/measurable-maps/metric.tex @@ -125,7 +125,7 @@ \begin{proposition} \label{proposition:measurable-simple-separable-norm} - Let $(X, \cm)$ be a measurable space, $(E, \norm{\cdot}_E)$ be a separable normed space, and $f: X \to E$, then the following are equivalent: + Let $(X, \cm)$ be a measurable space, $(E, \norm{\cdot}_E)$ be a separable normed vector space, and $f: X \to E$, then the following are equivalent: \begin{enumerate} \item $f$ is $(\cm, \cb_E)$-measurable. \item There exists simple functions $\seq{f_n}$ such that $\abs{f_n} \le \abs{f}$ for all $n \in \natp$, and $f_n \to f$ pointwise. diff --git a/src/measure/radon/c0.tex b/src/measure/radon/c0.tex index ff2114c..3adf422 100644 --- a/src/measure/radon/c0.tex +++ b/src/measure/radon/c0.tex @@ -26,7 +26,7 @@ \begin{definition}[Space of Finite Radon Measures] \label{definition:space-radon-measures} - Let $X$ be a LCH space and $E$ be a normed space over $K \in \RC$, then $M_R(X; E)$ is the \textbf{space of finite Radon measures} on $X$, which forms a vector space over $K$. + Let $X$ be a LCH space and $E$ be a normed vector space over $K \in \RC$, then $M_R(X; E)$ is the \textbf{space of finite Radon measures} on $X$, which forms a vector space over $K$. \end{definition} \begin{proof} Let $\mu, \nu \in M_R(X; E)$, then for any $A \in \cb_X$, $|\mu + \nu|(A) \le |\mu|(A) + |\nu|(A)$. Let $\eps > 0$, then by outer regularity and \autoref{proposition:radon-measurable-description}, there exists $K \subset A$ compact and $U \in \cn^o(A)$ such that $(|\mu| + |\nu|)(A \setminus K), (|\mu| + |\nu|)(U \setminus A) < \eps$. Therefore $|\mu + \nu|$ is regular on all Borel sets, and hence Radon. diff --git a/src/measure/radon/radon.tex b/src/measure/radon/radon.tex index 2027b06..8522405 100644 --- a/src/measure/radon/radon.tex +++ b/src/measure/radon/radon.tex @@ -150,7 +150,7 @@ \begin{proposition} \label{proposition:radon-cc-dense} - Let $X$ be a LCH space, $\mu: \cb_X \to [0, \infty]$ be a Radon measure, $E$ be a normed space, and $p \in [1, \infty)$, then $C_c(X; E)$ is dense in $L^p(X; E)$. + Let $X$ be a LCH space, $\mu: \cb_X \to [0, \infty]$ be a Radon measure, $E$ be a normed vector space, and $p \in [1, \infty)$, then $C_c(X; E)$ is dense in $L^p(X; E)$. \end{proposition} \begin{proof} By \autoref{proposition:lp-simple-dense}, $\Sigma(X, \cm; E) \cap L^p(X; E)$ is dense in $L^p(X; E)$. Using linearity, it is sufficient to approximate indicator functions of Borel sets with finite measure. diff --git a/src/measure/vector/variation.tex b/src/measure/vector/variation.tex index 160d394..905518d 100644 --- a/src/measure/vector/variation.tex +++ b/src/measure/vector/variation.tex @@ -3,7 +3,7 @@ \begin{definition}[Total Variation of Vector Measures] \label{definition:total-variation-vector} - Let $(X, \cm)$ be a measure space, $\mu$ be a vector measure taking values in a normed space $E$, and + Let $(X, \cm)$ be a measure space, $\mu$ be a vector measure taking values in a normed vector space $E$, and \[ |\mu|(A) = \sup\bracs{\sum_{j = 1}^n \norm{\mu(A_j)}_E \bigg | \seqf{A_j} \subset \cm, A = \bigsqcup_{j = 1}^n A_j} \] diff --git a/src/measure/vector/vector.tex b/src/measure/vector/vector.tex index f1c7e1d..3957531 100644 --- a/src/measure/vector/vector.tex +++ b/src/measure/vector/vector.tex @@ -3,7 +3,7 @@ \begin{definition}[Vector Measure] \label{definition:vector-measure} - Let $(X, \cm)$ be a measurable space, $E$ be a normed space over $K \in \RC$, and $\mu: \cm \to E$, then $\mu$ is a \textbf{vector measure} if: + Let $(X, \cm)$ be a measurable space, $E$ be a normed vector space over $K \in \RC$, and $\mu: \cm \to E$, then $\mu$ is a \textbf{vector measure} if: \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. @@ -13,7 +13,7 @@ \begin{proposition} \label{proposition:vector-measure-bounded} - Let $(X, \cm)$ be a measurable space, $E$ be a normed space over $K \in \RC$, and $\mu: \cm \to E$ be a vector measure, then + Let $(X, \cm)$ be a measurable space, $E$ be a normed vector space over $K \in \RC$, and $\mu: \cm \to E$ be a vector measure, then \[ \sup_{A \in \cm}\norm{\mu(A)}_E < \infty \]