Added barreled spaces.

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}
+

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]

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 $M = \bracs{0}$.
 

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}

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

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}