Updated the notation for convex and circled hulls.

preamble.sty

@@ -224,3 +224,8 @@
 
 % Real or Complex Numbers
 \newcommand{\RC}{\bracs{\real, \complex}}
+
+% Convex Stuff
+\newcommand{\conv}{\text{Conv}}
+\newcommand{\aconv}{\text{AbsConv}}
+

src/fa/lc/convex.tex

@@ -21,7 +21,7 @@
 \label{definition:convex-circled-hull}
     Let $E$ be a vector space over $K \in \RC$ and $A \subset E$, then the set
     \[
-    \Gamma(A) = \bracs{\sum_{j = 1}^n t_j x_j \bigg | \seqf{t_j} \subset K, \seqf{x_j} \subset E, \sum_{j = 1}^n |t_j| \le 1 }
+    \aconv(A) = \bracs{\sum_{j = 1}^n t_j x_j \bigg | \seqf{t_j} \subset K, \seqf{x_j} \subset E, \sum_{j = 1}^n |t_j| \le 1 }
     \]
 
     is the \textbf{convex circled hull} of $A$. 

src/fa/lc/spaces-of-linear.tex

@@ -3,21 +3,16 @@
 
 \begin{proposition}
 \label{proposition:lc-spaces-linear-map}
-    Let $T$ be a set, $\sigma \subset 2^T$ be an ideal, and $E$ be a locally convex space over $K$. For each $S \in \sigma$ and continuous seminorm $\rho: E \to [0, \infty)$, let
+    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
-    \[
-    \rho_S: E^T \to [0, \infty] \quad f \mapsto \sup_{x \in S}\rho(f(x))
-    \]
-
-    then the the $\sigma$-uniform topology on $E^T$ is defined by distances of the form
-    \[
-    d_{S, \rho}: E^T \times E^T \to [0, \infty] \quad (f, g) \mapsto \rho_S(f - g)
-    \]
-
-    In particular, if $\cf \subset E^T$ is 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}
     
+    For each $S \in \sigma$ and continuous seminorm $\rho: E \to [0, \infty)$, let
+    \[
+    \rho_S: E^T \to [0, \infty] \quad f \mapsto \sup_{x \in S}\rho(f(x))
+    \]
+
     then the $\sigma$-uniform topology on $\cf$ is induced by seminorms of the form $\rho_S$, where $\rho$ is a continuous seminorm on $E$, and $S \in \sigma$. In which case, the $\sigma$-uniform topology on $\cf$ is locally convex. 
 \end{proposition}
 \begin{proof}

src/fa/lc/tensor.tex

@@ -19,7 +19,7 @@
         \item $E \otimes_\pi F$ is the linear span of $\iota(E \times F)$. 
         \item For any $U \subset E$ and $V \subset F$, let $U \otimes V = \bracs{u \otimes v|u \in U, v \in V}$, then the convex circled hulls
         \[
-        \fB = \bracsn{\Gamma(U \otimes V)| U \in \cn_E(0), V \in \cn_F(0)}
+        \fB = \bracsn{\aconv(U \otimes V)| U \in \cn_E(0), V \in \cn_F(0)}
         \]
 
         is a fundamental system of neighbourhoods at $0$ for $E \otimes_\pi F$.
@@ -42,9 +42,9 @@
 
     (5): By (4) of the \hyperref[tensor product]{definition:tensor-product}. 
 
-    (6): Let $U \in \cn_E(0)$ and $V \in \cn_F(0)$ be balanced. For any $\sum_{j = 1}^n x_j \otimes y_j \in E \otimes_\pi F$, then there exists $\lambda > 0$ such that $\seqf{x_j} \subset \lambda U$ and $\seqf{y_j} \subset \lambda V$. In which case, $\sum_{j = 1}^n x_j \otimes y_j \subset \lambda \Gamma (U \otimes V)$, so $\fB$ is a collection of convex, circled, and radial sets. Since $\fB$ defines a locally convex topology that satisfies (1) and (2), $\mathcal{S}$ contains the topology defined by $\fB$. 
+    (6): Let $U \in \cn_E(0)$ and $V \in \cn_F(0)$ be balanced. For any $\sum_{j = 1}^n x_j \otimes y_j \in E \otimes_\pi F$, then there exists $\lambda > 0$ such that $\seqf{x_j} \subset \lambda U$ and $\seqf{y_j} \subset \lambda V$. In which case, $\sum_{j = 1}^n x_j \otimes y_j \subset \lambda \aconv (U \otimes V)$, so $\fB$ is a collection of convex, circled, and radial sets. Since $\fB$ defines a locally convex topology that satisfies (1) and (2), $\mathcal{S}$ contains the topology defined by $\fB$. 
 
-    On the other hand, for any convex and circled set $W \in \cn_{E \otimes_\pi F}(0)$, there exists $U \in \cn_E(0)$ and $V \in \cn_F(0)$ such that $U \otimes V \subset W$. In which case, $W \supset \Gamma(U \otimes V)$, so $\fB$ is a fundamental system of neighbourhoods at $0$ for $E \otimes_\pi F$. 
+    On the other hand, for any convex and circled set $W \in \cn_{E \otimes_\pi F}(0)$, there exists $U \in \cn_E(0)$ and $V \in \cn_F(0)$ such that $U \otimes V \subset W$. In which case, $W \supset \aconv(U \otimes V)$, so $\fB$ is a fundamental system of neighbourhoods at $0$ for $E \otimes_\pi F$. 
 \end{proof}
 
 \begin{remark}
@@ -64,7 +64,7 @@
     then
     \begin{enumerate}
         \item $\rho$ is a continuous seminorm on $E \otimes_\pi F$.
-        \item $\rho$ is the gauge of $\Gamma(U \otimes V)$. 
+        \item $\rho$ is the gauge of $\aconv(U \otimes V)$. 
         \item For any $x \in E$ and $y \in F$, $\rho(x \otimes y) = p(x)q(Y)$. 
         \item $\rho$ is a norm if and only if $[\cdot]_U$ and $[\cdot]_V$ are norms. 
     \end{enumerate}
@@ -100,20 +100,20 @@
 
     so $\rho$ satisfies the triangle inequality. 
 
-    (2): Let $z \in \Gamma(U \otimes V)$, then there exists $\seqf{(x_j, y_j)} \subset U \times V$ and $\seqf{\lambda_j} \subset K$ such that $\sum_{j = 1}^n |\lambda_j| \le 1$ and $z = \sum_{j = 1}^n \lambda x_j \otimes y_j$. In which case, 
+    (2): Let $z \in \aconv(U \otimes V)$, then there exists $\seqf{(x_j, y_j)} \subset U \times V$ and $\seqf{\lambda_j} \subset K$ such that $\sum_{j = 1}^n |\lambda_j| \le 1$ and $z = \sum_{j = 1}^n \lambda x_j \otimes y_j$. In which case, 
     \begin{align*}
         \rho(z) &\le \sum_{j = 1}^n p(\lambda x_j)q(y_j) = \sum_{j = 1}^n |\lambda_j|p(x_j)q(y_j) \\
         &< \sum_{j = 1}^n |\lambda_j| \le 1
     \end{align*}
 
-    so $\Gamma(U \otimes V) \subset \bracs{\rho < 1}$.
+    so $\aconv(U \otimes V) \subset \bracs{\rho < 1}$.
 
     Let $z \in \bracs{\rho < 1}$, then there exists $\seqf{(x_j, y_j)} \subset E \times F$ such that $z = \sum_{j = 1}^nx_j \otimes y_j$ and $\sum_{j = 1}^n p(x_j)q(x_j) < 1$. Let $\eps > 0$ such that $\sum_{j = 1}^n(p(x_j) + \eps)(q(x_j) + \eps) < 1$, then
     \[
-    z = \sum_{j = 1}^n (p(x_j) + \eps)(q(x_j) + \eps) \cdot \underbrace{\frac{x_j}{p(x_j) + \eps}}_{\in \bracs{p < 1} = U} \otimes \underbrace{\frac{y_j}{q(x_j) + \eps}}_{\in \bracs{q < 1} = V} \in \Gamma(U \otimes V)
+    z = \sum_{j = 1}^n (p(x_j) + \eps)(q(x_j) + \eps) \cdot \underbrace{\frac{x_j}{p(x_j) + \eps}}_{\in \bracs{p < 1} = U} \otimes \underbrace{\frac{y_j}{q(x_j) + \eps}}_{\in \bracs{q < 1} = V} \in \aconv(U \otimes V)
     \]
 
-    and $\Gamma(U \otimes V) \supset \bracs{\rho < 1}$.
+    and $\aconv(U \otimes V) \supset \bracs{\rho < 1}$.
     
     (3): Let $x \in U$ and $y \in V$. By the \hyperref[Hahn-Banach Theorem]{proposition:hahn-banach-utility}, there exists $\phi \in E^*$ and $\psi \in F^*$ such that $\dpn{x, \phi}{E} = p(x)$, $\dpn{y, \psi}{F} = q(x)$, $|\phi| \le p$, and $|\psi| \le q$. By (U1) of the \hyperref[projective tensor product]{definition:projective-tensor-product}, there exists $\Phi \in (E \otimes_\pi F)^*$ such that the following diagram commutes
     \[

src/fa/notation.tex

@@ -20,7 +20,7 @@
     $\widehat{E}$ & Hausdorff completion of TVS $E$. & \autoref{definition:tvs-completion} \\
     % ---- Locally Convex ----
     $\mathrm{Conv}(A)$ & Convex hull of $A$. & \autoref{definition:convex-hull} \\
-    $\Gamma(A)$ & Convex circled hull of $A$. & \autoref{definition:convex-circled-hull} \\
+    $\aconv(A)$ & Convex circled hull of $A$. & \autoref{definition:convex-circled-hull} \\
     $[\cdot]_A$ & Gauge of a radial set $A$. & \autoref{definition:gauge} \\
     $\rho_M$ & Quotient of seminorm $\rho$ by subspace $M$. & \autoref{definition:quotient-norm} \\
     $E \otimes_\pi F$ & Projective tensor product of $E$ and $F$. & \autoref{definition:projective-tensor-product} \\