From 2bacd9b3700ff9ebce99b47e57dd123219af0618 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Tue, 27 Jan 2026 14:10:21 -0500 Subject: [PATCH] Updated Schaefer & Wolff citations. --- src/fa/tvs/bounded.tex | 2 +- src/fa/tvs/completion.tex | 2 +- src/fa/tvs/definition.tex | 6 +++--- src/fa/tvs/metric.tex | 4 ++-- src/fa/tvs/spaces-of-linear.tex | 2 +- 5 files changed, 8 insertions(+), 8 deletions(-) diff --git a/src/fa/tvs/bounded.tex b/src/fa/tvs/bounded.tex index 815a428..4a3e48c 100644 --- a/src/fa/tvs/bounded.tex +++ b/src/fa/tvs/bounded.tex @@ -6,7 +6,7 @@ Let $E$ be a TVS over $K \in \RC$ and $B \subset E$, then $B$ is \textbf{bounded} if for every $U \in \cn(0)$, there exists $\lambda \in K$ such that $\lambda U \supset B$. The collection $B(E)$ is the set of all bounded sets of $E$. \end{definition} -\begin{proposition}[{{\cite[1.5.1]{SchaeferWolff}}}] +\begin{proposition}[{{\cite[I.5.1]{SchaeferWolff}}}] \label{proposition:bounded-operations} Let $E$ be a TVS over $K \in \RC$ and $A, B \in B(E)$, then the following sets are bounded: \begin{enumerate} diff --git a/src/fa/tvs/completion.tex b/src/fa/tvs/completion.tex index 77ec3a3..1d9fbf7 100644 --- a/src/fa/tvs/completion.tex +++ b/src/fa/tvs/completion.tex @@ -29,6 +29,6 @@ By continuity and the density of $\iota(E)$ in $E$, $\wh E$ with these operations forms a TVS, and $T$ is linear. \end{proof} -\begin{remark}[{{\cite[Section 1.1]{SchaeferWolff}}}] +\begin{remark}[{{\cite[Section I.1]{SchaeferWolff}}}] The Hausdorff completion works in general with arbitrary valuated fields. Though the completion yields a TVS over the completion of the field, the field need not to be complete. \end{remark} diff --git a/src/fa/tvs/definition.tex b/src/fa/tvs/definition.tex index 4291160..b455aa3 100644 --- a/src/fa/tvs/definition.tex +++ b/src/fa/tvs/definition.tex @@ -43,7 +43,7 @@ \end{proof} -\begin{proposition}[{{\cite[1.1.4]{SchaeferWolff}}}] +\begin{proposition}[{{\cite[I.1.4]{SchaeferWolff}}}] \label{proposition:tvs-uniform} Let $E$ be a TVS over $K \in \bracs{\real, \complex}$, then: \begin{enumerate} @@ -85,7 +85,7 @@ \] \end{proof} -\begin{proposition}[{{\cite[1.1.1]{SchaeferWolff}}}] +\begin{proposition}[{{\cite[I.1.1]{SchaeferWolff}}}] \label{proposition:tvs-set-operations} Let $E$ be a TVS over $K \in \RC$ and $A, B \subset E$, then: \begin{enumerate} @@ -148,7 +148,7 @@ Let $U \in \cn(0)$ be closed, then there exists a balanced neighbourhood $V \in \cn^o(0)$ such that $V \subset U$. In which case, for any $\lambda \in K$ with $0 < \abs{\lambda} \le 1$, $\lambda \overline{V} = \overline{\lambda V} \subset \overline{V}$ by (TVS2). Therefore $\overline{V} \subset U$ is balanced as well. \end{proof} -\begin{proposition}[{{\cite[1.2]{SchaeferWolff}}}] +\begin{proposition}[{{\cite[I.1.2]{SchaeferWolff}}}] \label{proposition:tvs-0-neighbourhood-base} Let $E$ be a vector space over $K \in \RC$, and $\topo$ be a vector space topology on $E$, then there exists a fundamental system of neighbourhoods $\fB \subset \cn_E(0)$ such that: \begin{enumerate} diff --git a/src/fa/tvs/metric.tex b/src/fa/tvs/metric.tex index ce5b60c..2a43946 100644 --- a/src/fa/tvs/metric.tex +++ b/src/fa/tvs/metric.tex @@ -71,7 +71,7 @@ $(4) \Rightarrow (1)$: By \ref{definition:tvs-pseudonorm-topology}, for each $r > 0$, $\rho^{-1}([0, r)) \in \cn_E(0)$. Thus for any $x, y \in E$, if $x - y \in \rho^{-1}([0, r))$, then $\abs{\rho(x) - \rho(y)} \le r$. Therefore $\rho \in UC(E; [0, \infty))$. \end{proof} -\begin{lemma}[{{\cite[Theorem 1.6.1]{SchaeferWolff}}}] +\begin{lemma}[{{\cite[Theorem I.6.1]{SchaeferWolff}}}] \label{lemma:tvs-sequence-pseudonorm} Let $E$ be a vector space over $K \in \RC$, $\seq{U_n} \subset 2^E$ such that \begin{enumerate} @@ -143,7 +143,7 @@ Let $E$ be a TVS over $K \in \RC$, then $E$ is \textbf{locally bounded} if there exists $U \in \cn^o(0)$ bounded. \end{definition} -\begin{proposition}[{{\cite[1.6.2]{SchaeferWolff}}}] +\begin{proposition}[{{\cite[I.6.2]{SchaeferWolff}}}] \label{proposition:locally-bounded-metrisable} Let $E$ be a locally bounded TVS over $K \in \RC$, then there exists a pseudonorm $\rho: E \to [0, \infty)$ that induces the topology on $E$. \end{proposition} diff --git a/src/fa/tvs/spaces-of-linear.tex b/src/fa/tvs/spaces-of-linear.tex index 3004060..02e0e1f 100644 --- a/src/fa/tvs/spaces-of-linear.tex +++ b/src/fa/tvs/spaces-of-linear.tex @@ -1,7 +1,7 @@ \section{Vector-Valued Function Spaces} \label{section:spaces-linear-map} -\begin{proposition}[{{\cite[3.3.1]{SchaeferWolff}}}] +\begin{proposition}[{{\cite[III.3.1]{SchaeferWolff}}}] \label{proposition:tvs-set-uniformity} Let $T$ be a set, $\mathfrak{S} \subset 2^T$ be an upward-directed system of sets, $F$ be a TVS over $K \in \RC$, then \begin{enumerate}