Updated nuclear spaces.

src/fa/lc/nuclear-space.tex

@@ -54,7 +54,7 @@
 
 \begin{theorem}
 \label{theorem:nuclear-lp}
-    Let $E$ be a nuclear space over $K \in \RC$, $U \in \cn_E(0)$, and $p \in [1, \infty]$, then there exists $V \in \cn_E(0)$ with $V \subset U$ such that $\wh E_V$ is isomorphic to a subspace of $l^p(\natp; K)$ with equal norms.  
+    Let $E$ be a nuclear space over $K \in \RC$, $U \in \cn_E(0)$, and $p \in [1, \infty]$, then there exists $V \in \cn_E(0)$ with $V \subset U$ such that $\wh E_V$ is isometrically isomorphic to a subspace of $l^p(\natp; K)$.
 \end{theorem}
 \begin{proof}[Proof, {{\cite[III.7.3]{SchaeferWolff}}}. ]
     Assume without loss of generality that $U$ is convex and circled, and the canonical projection $\pi_U: E \to \wh E_U$ is nuclear. In which case, there exists an equicontinuous sequence $\seq{\phi_n} \subset E^*$, $\seq{y_n} \subset B_{\wh E_U}(0, 1)$, and $\seq{\lambda_n} \subset K$ such that
@@ -81,5 +81,9 @@
     Finally, let $V = T^{-1}(B_{l^p(\natp; K)})$, then $V \subset U$, and $\wh E_V$ is isomorphic to $\ol{T(E)}$, with equal norms. 
 \end{proof}
 
+\begin{corollary}
+\label{corollary:complete-nuclear-projective-limit}
+    Let $E$ be a complete nuclear space over $K \in \RC$, then $E$ is a projective limit of Hilbert spaces over $K$. For any Fréchet space $F$, $F$ is nuclear if and only if it is the projective limit of a sequence $\seq{H_n}$ of Hilbert spaces such that the mapping $H_m \to H_n$ is nuclear for all $1 \le m < n < \infty$. 
+\end{corollary}
 
 

src/fa/lc/projective.tex

@@ -23,7 +23,7 @@
     By (U) of \autoref{definition:tvs-projective-limit} and \autoref{definition:tvs-initial}, $E$ is equipped with the projective topology generated by the projection maps $E \to E_i$. By \autoref{proposition:lc-projective-topology}, $E$ is locally convex.
 \end{proof}
 
-\begin{proposition}[{{\cite[II.5.4]{SchaeferWolff}}}]
+\begin{proposition}
 \label{proposition:complete-lc-projective-limit}
     Let $E$ be a separated complete locally convex space over $K \in \RC$, $\mathcal{B} \subset \cn_E(0)$ be a fundamental system of neighbourhoods consisting of convex, circled, and radial sets, directed under inclusion.
 
@@ -44,7 +44,7 @@
 
     \end{enumerate}
 \end{proposition}
-\begin{proof}
+\begin{proof}[Proof, {{\cite[II.5.4]{SchaeferWolff}}}]
     (1): Since $V \supset U$, $[\cdot]_V \ge [\cdot]_U$, so $M_V \supset M_U$. Thus $\ker(\pi_V) \supset M_U$. By (U) of the \hyperref[quotient]{definition:tvs-quotient}, $\pi_V$ factors through $E_U$ as $\pi^U_V$, so $\pi^U_V \in L(E_U; E_V)$.
 
     (2): Since $\mathcal{B}$ is a fundamental system of neighbourhoods, it is downward-directed under inclusion. For any $U, V, W \in \mathcal{B}$ with $U \subset V \subset W$, $M_U \supset M_V \supset M_W$. Thus $\pi^U_W = \pi^V_W \circ \pi^U_V$.