\section{The Approximation Property}
\label{section:approximation-property}

\begin{definition}[Approximation Property]
\label{definition:approximation-property}
    Let $E$ be a separated locally convex space over $K \in \RC$, then the following are equivalent:
    \begin{enumerate}
        \item The closure of $E^* \otimes E$ in $L_c(E; E)$ contains the identity map.
        \item $E^* \otimes E$ is dense in $L_c(E; E)$. 
        \item For each locally convex space $F$ over $K$, $E^* \otimes F$ is dense in $L_c(E; F)$. 
        \item For each locally convex space $F$ over $K$, $F^* \otimes E$ is dense in $L_c(F; E)$. 
    \end{enumerate}

    If the above holds, then $E$ has the \textbf{approximation property}. 
\end{definition}
\begin{proof}
    (1) $\Rightarrow$ (2): Let $T \in L_c(E; E)$ and $A \subset E$ be precompact, then $T(A)$ is also precompact by \autoref{proposition:totally-bounded-image}. Let $U \in \cn_E(0)$, then there exists $S \in E^* \otimes E$ such that $Sx - x \in U$ for all $x \in T(A)$. In which case, $STx - Tx \in U$ for all $x \in A$. 

    (1) $\Rightarrow$ (3): Let $T \in L_c(E; F)$ and $A \subset E$ be precompact, and $U \in \cn_F(0)$, then there exists $S \in E^* \otimes E$ such that $Sx - x \in T^{-1}(U)$ for all $x \in A$. In which case, $TS \in E^* \otimes F$ and $TSx - Tx \in U$ for all $x \in A$. 

    (1) $\Rightarrow$ (4): Let $T \in L_c(F; E)$ and $A \subset F$ be precompact, then $T(A)$ is also precompact. Let $U \in \cn_E(0)$, then there exists $S \in E^* \otimes E$ such that $Sx - x \in U$ for all $x \in T(A)$. Thus $STx - Tx \in U$ for all $x \in A$.  
\end{proof}

\begin{proposition}
\label{proposition:approximation-property-associated}
    Let $E$ be a locally convex space over $K \in \RC$. If there exists a fundamental system of convex and circled neighbourhoods $\fB \subset \cn_E(0)$ such that for each $V \in \fB$, $\wh E_V$ has the approximation property, then $E$ has the approximation property. 
\end{proposition}
\begin{proof}
    Let $V \in \fB$, $\pi_V: E \to \wh E_V$ be the canonical projection, and $A \subset E$ be precompact, then $\pi_V(A)$ is precompact as well. Since $\wh E_V$ has the approximation property, there exists $T \in E_V^* \otimes \wh E_V$ such that $Tx - x \in \pi_V(V)$ for all $x \in \pi_V(A)$. As $E_V$ is dense in $\wh E_V$, there exists $S \in E_V^* \otimes E_V$ such that $Sx - Tx \in \pi_V(V)$ for all $x \in \pi_V(A)$. In which case, $Sx - x \in 2\pi_V(V)$ for all $x \in \pi_V(A)$, and $S \circ \pi_V(x) - \pi_V(x) \in 2\pi_V(V)$. 
    
    Write $S = \sum_{j = 1}^n \phi_j \otimes y_j$. For each $1 \le j \le n$, choose any representative $x_j \in \pi_V^{-1}(y_j)$, then for any $x \in A$, 
    \[
        \pi_V \braks{x - \sum_{j = 1}^n x_j\dpn{x, \phi_j \circ \pi_V}{E}} = S \circ \pi_V(x) - \pi_V(x) \in -2\pi_V(V) = 2\pi_V(V)
    \]

    Finally, since $\ker(\pi_V) = \bigcap_{\lambda > 0}\lambda V \subset V$, $x - \sum_{j = 1}^n x_j\dpn{x, \phi_j \circ \pi_V}{E} \in -3V = 3V$. Therefore if $R = \sum_{j = 1}^n (\phi_j \circ \pi_V) \otimes x_j \in E^* \otimes E$, then $Rx - x \in 3V$. 
\end{proof}

\begin{corollary}
\label{corollary:approximation-property-hilbert}
    Every subspace of a product of Hilbert spaces has the approximation property. Every subspace of a projective limit of Hilbert spaces has the approximation property. 
\end{corollary}

\begin{lemma}
\label{lemma:compact-operator-topology-banach-dual}
    Let $E, F$ be Banach spaces over $K \in \RC$ and $\phi \in L_c(E; F)^*$, then there exists $\seq{x_n} \subset E$ and $\seq{\psi_n} \subset F^*$ such that:
    \begin{enumerate}
        \item $\limv{n}x_n = 0$. 
        \item $\sum_{n \in \natp}\norm{\psi_n}_{F^*} < \infty$. 
        \item For each $T \in L(E; F)$, $\dpn{T, \phi}{L_c(E; F)} = \sum_{n = 1}^\infty \dpn{Tx_n, \psi_n}{F}$. 
    \end{enumerate}
\end{lemma}
\begin{proof}
    Since $\phi \in L_c(E; F)^*$, there exists $A \subset E$ compact and $\alpha > 0$ such that $|\dpn{T, \phi}{L_c(E; F)}| \le \alpha\sup_{x \in A}\norm{Tx}_F$ for all $T \in L(E; F)$. After rescaling $A$, assume without loss of generality that $\alpha = 1$, so that $|\dpn{T, \phi}{L_c(E; F)}| \le \sup_{x \in A}\norm{Tx}_F$ for all $T \in L(E; F)$. 

    By \autoref{lemma:compact-null-auxiliary}, there exists $\seq{x_n} \subset E$ with $\limv{n}x_n = 0$ and $A \subset \ol{\conv}(\seq{x_n})$. Since $E$ is complete, \hyperref[Mazur's Theorem]{theorem:convex-hull-complete} implies that $\ol{\conv}(\seq{x_n})$ is compact as well. Thus for each $T \in L(E; F)$, 
    \[
    T(A) \subset T(\ol{\conv}(\seq{x_n})) = \ol{\conv}(T(\seq{x_n}))
    \]

    In particular, 
    \[
    |\dpn{T, \phi}{L_c(E; F)}| \le \sup_{x \in A}\norm{Tx}_F \le \sup_{n \in \natp}\norm{Tx_n}_F
    \]

    As $\seq{x_n}$ is a null sequence in $E$, $\seq{Tx_n} \in c_0(\natp; F)$ for each $T \in L(E; F)$. Let $L = \bracs{\seq{Tx_n}|T \in L(E; F)}$, then $L$ is a subspace of $c_0(\natp; F)$. By the above estimate, $\phi$ factors through $L$ as follows: 
    \[
    \xymatrix{
    L_c(E; F) \ar@{->}[rd]_{\phi} \ar@{->}[r] & L \ar@{->}[d]^{\widehat \phi} \\
    & K
    }
    \]

    The \hyperref[Hahn-Banach Theorem]{theorem:hahn-banach} then yields an extension $\Phi$ of $\wh \phi$ as shown below:
    \[
    \xymatrix{
    L_c(E; F) \ar@{->}[rd]_{\phi} \ar@{->}[r] & L \ar@{->}[d]^{\widehat \phi} \ar@{->}[r] & c_0(\mathbb{N}^+; F) \ar@{->}[ld]^{\Phi} \\
    & K & 
    }
    \]

    By \autoref{theorem:c0-sum-dual}, there exists $\seq{\psi_n} \in l^1(\natp; F^*)$ such that for each $y \in c_0(\natp; F)$, $\dpn{y, \Phi}{c_0(\natp; F)} = \sum_{n = 1}^\infty \dpn{y_n, \psi_n}{F}$. In particular, for each $T \in L(E; F)$, 
    \begin{align*}
        \dpn{T, \phi}{L_c(E; F)} &= \dpn{\seq{Tx_n}, \wh \phi}{L} = \dpn{\seq{Tx_n}, \Phi}{c_0(\natp; F)} \\
        &= \sum_{n = 1}^\infty \dpn{Tx_n, \psi_n}{F}
    \end{align*}
\end{proof}

\begin{remark}
\label{remark:compact-operator-topology-banach-dual}
    In \autoref{lemma:compact-operator-topology-banach-dual}, I would like to say that the mapping from $E \wh \otimes_\pi F^*$ to $L_c(E; F)^*$ defined by (3) is surjective. However, it does not seem right to me that this mapping is continuous at all. As such, I decided against mentioning the projective completion for this lemma. 
\end{remark}


\begin{theorem}
\label{theorem:approximation-property-dual}
    Let $E$ be a Banach space over $K \in \RC$, then the following are equivalent:
    \begin{enumerate}
        \item $E$ has the approximation property. 
        \item For any Banach space $F$, the closure of $F^* \otimes E$ in $L(F; E)$ is $\mathcal{K}(F; E)$.
        \item For any Banach space $F$, the canonical map $F^* \wh \otimes_\pi E \to L(F; E)$ is injective. 
        \item The canonical map $E^* \wh \otimes_\pi E \to L(E; E)$ is injective.
    \end{enumerate}

    and the following are equivalent:
    \begin{enumerate}[label=(\arabic**)]
        \item $E^*$ has the approximation property. 
        \item For any Banach space $F$, the closure of $E^* \otimes F$ in $L(E; F)$ is $\mathcal{K}(E; F)$.
    \end{enumerate}
\end{theorem}
\begin{proof}[Proof, {{\cite[Theorem III.9.5]{SchaeferWolff}}} and {{\cite[Proposition 4.6]{RyanTensor}}}. ]
    (1) $\Rightarrow$ (2): Let $T \in \mathcal{K}(F; E)$, then $T(B_F(0, 1))$ is precompact in $E$. Thus for any $\eps > 0$, there exists $S \in E^* \otimes E$ such that $\norm{Sy - y}_{E} < \eps$ for all $y \in T(B_F(0, 1))$. In which case, $\norm{STx - Tx}_{E} < \eps$ for all $x \in B_F(0, 1)$. Therefore $ST \in F^* \otimes E$ with $\norm{ST - T}_{L(F; E)} \le \eps$. 

    (2) $\Rightarrow$ (1): Let $A \subset E$ be compact and $\eps > 0$. By \autoref{lemma:compact-null-auxiliary}, there exists a convex, circled, and compact set $B \subset E$ such that $A$ is compact as a subset of $E_B$. 

    Since $B$ is compact, the inclusion $E_B \to E$ is compact. By (2) applied to the inclusion map, there exists $T \in E_B^* \otimes E$ such that $\norm{Tx - x}_{E} \le \eps \norm{x}_{E_B}$ for all $x \in E_B$. 

    Write $T = \sum_{j = 1}^n \phi_j \otimes x_j$, then as $A$ is compact in $E_B$, \hyperref[Goldstine's Theorem]{theorem:goldstine-weak} and the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli} imply that there exists $\seqf{\psi_j} \subset E^*$ such that $|\dpn{x, \phi_j - \psi_j}{E_B}| \le \eps/\sum_{j = 1}^n \norm{x_j}_E$ for all $x \in A$ and $1 \le j \le n$. In which case, 
    \begin{align*}
        \norm{x - \sum_{j = 1}^n x_j\dpn{x, \psi_j}{E}}_E &\le \norm{x - \sum_{j = 1}^n x_j\dpn{x, \phi_j}{E_B}}_E \\
        &+  \sum_{j = 1}^n \norm{x_j}_E|\dpn{x, \phi_j - \psi_j}{E_B}| \\
        &\le \eps\norm{x}_{E_B} + \eps \le \eps\braks{1 + \sup_{x \in A}\norm{x}_{E_B}}
    \end{align*}

    for all $x \in A$. Therefore $S = \sum_{j = 1}^n \psi_j \otimes x_j \in E^* \otimes E$ with $\norm{Sx - x}_{E} \le \eps\braks{1 + \sup_{x \in A}\norm{x}_{E_B}}$ for all $x \in A$, and $E$ has the approximation property. 

    (1) $\Rightarrow$ (3): Let $T \in F^* \wh \otimes_\pi E$ such that $Tx = 0$ for all $x \in F$. By \autoref{theorem:metrisable-tensor-product}, there exists $\seq{\phi_n} \subset F^*$ and $\seq{x_n} \subset E$ such that $T = \sum_{n = 1}^\infty \phi_n \otimes x_n$. $\sum_{n \in \natp}\norm{\phi_n}_{F^*}\norm{x_n}_E < \infty$, $\limv{n}x_n = 0$, and $\sum_{n \in \natp}\norm{\phi_n}_{F^*} < \infty$. 

    Let $A$ be the closure of $\seq{x_n}$, then as $\seq{x_n}$ is a null sequence, $A$ is compact. Let $S \in L(E; F^{**}) = (F^* \wh \otimes_\pi E)^*$ and $\eps > 0$, then there exists $R \in E^{*} \otimes F^{**}$ such that $\norm{Rx - Sx}_{F^{**}} \le \eps$ for all $x \in A$. Write $R = \sum_{k = 1}^m \psi_k \otimes y_k$, then by \autoref{proposition:projective-tensor-product-dual}, 
    \begin{align*}
        \dpn{T, R}{F^* \wh \otimes_\pi E} &= \sum_{n = 1}^\infty \dpn{\phi_n, Rx_n}{F^*} = \sum_{n = 1}^\infty \angles{\phi_n, \sum_{k = 1}^m y_k \dpn{x_n, \psi_k}{E}}_{F^*} \\
        &= \sum_{k = 1}^m \sum_{n = 1}^\infty \dpn{\phi_n, y_k}{F^*} \dpn{x_n, \psi_k}{E}
    \end{align*}

    Since $\sum_{n \in \natp}\norm{\phi_n}_{F^*} < \infty$, assume without loss of generality that $\bracsn{y_k}_1^m \subset F$ with \hyperref[Goldstine's Theorem]{theorem:goldstine-weak}. This allows rewriting
    \[
    \dpn{T, R}{F^* \wh \otimes_\pi E} = \sum_{k = 1}^m \dpn{Ty_k, \psi_k}{E} = 0
    \]

    so
    \[
    |\dpn{T, S}{F^* \wh \otimes_\pi E}| \le |\dpn{T, R}{F^* \wh \otimes_\pi E}| + \eps \sum_{n \in \natp}\norm{\phi_n}_{F^*} = \eps \sum_{n \in \natp}\norm{\phi_n}_{F^*}
    \]

    As the above holds for all $\eps > 0$, $\dpn{T, S}{F^* \wh \otimes_\pi E} = 0$. Therefore $T = 0$ as an element of $F^* \wh \otimes_\pi E$. 


    $\neg$ (1) $\Rightarrow$ $\neg$ (4): Suppose that $E$ suffers from a lack of the approximation property, then $\text{Id}$ is not in the closure of $E^* \otimes E$ in $L_c(E; E)$. By the \hyperref[Hahn-Banach Theorem]{proposition:hahn-banach-utility}, there exists $\phi \in L_c(E; E)^*$ such that $\dpn{\text{Id}, \phi}{L_c(E; E)} = 1$, but $\dpn{T, \phi}{L_c(E; E)} = 0$ for all $T \in E^* \otimes E$. 

    By \autoref{lemma:compact-operator-topology-banach-dual}, there exists a null sequence $\seq{x_n} \subset E$ and $\seq{\psi_n} \in l^1(\natp; E^*)$ such that for each $T \in L(E; E)$, 
    \[
    \dpn{T, \phi}{L_c(E; E)} = \sum_{n = 1}^\infty \dpn{Tx_n, \psi_n}{E}
    \]

    In particular, for any $x \in E$ and $\eta \in E^*$, 
    \begin{align*}
        0 &= \dpn{\eta \otimes x, \phi}{L_c(E; E)} = \sum_{n = 1}^\infty \dpn{x_n, \eta}{E} \dpn{x, \psi_n}{E} \\
        &= \angles{\sum_{n = 1}^\infty x_n\dpn{x, \psi_n}{E}, \eta}_E
    \end{align*}

    By the \hyperref[Hahn-Banach Theorem]{proposition:hahn-banach-utility}, $\sum_{n = 1}^\infty x_n \dpn{x, \psi_n}{E} = 0$. Thus $\sum_{n = 1}^\infty x_n \dpn{x, \psi_n}{E} =0 $ for all $x \in E$. 

    As $\seq{x_n}$ is a null sequence and $\seq{\psi_n} \in l^1(\natp; E^*)$, $\sum_{n \in \natp}\norm{x_n}_E\norm{\psi_n}_{E^*} < \infty$. This yields that $\sum_{n = 1}^\infty \psi_n \otimes x_n \in E^* \wh \otimes_\pi E$ with $\braks{\sum_{n = 1}^\infty \psi_n \otimes x_n} x = 0$ for all $x \in E$. 

    However, since $1 = \dpn{\text{Id}, \phi}{L_c(E; E)} = \sum_{n = 1}^\infty \dpn{x_n, \psi_n}{E}$, $ \sum_{n = 1}^\infty \psi_n \otimes x_n \ne 0$ as an element of $E^* \wh \otimes_\pi E$. Therefore the canonical mapping from $E^* \wh \otimes_\pi E$ to $L(E; E)$ is not injective. 


    (1*) $\Rightarrow$ (2*): Let $T \in \mathcal{K}(E; F)$ be compact, then $T^* \in \mathcal{K}(F^*; E^*)$ is compact by \hyperref[Schauder's Theorem]{theorem:compact-adjoint}, and $T^*(B_{F^*}(0, 1))$ is relatively compact.
    
    Let $\eps > 0$, then since $E^*$ has the approximation property, there exists $S \in E^{**} \otimes E^*$ such that $\norm{S\phi - \phi}_{E^*} \le \eps$ for all $\phi \in T^*(B_{F^*}(0, 1))$. 
    By \hyperref[Gantmacher's Theorem]{theorem:weakly-compact-biadjoint}, $T^{**}(E^{**}) \subset F$. Thus $T^{**}S^* \in E^{***} \otimes F \subset L(E, F)$. For any $x \in E$ and $\phi \in B_{F^*}(0, 1)$, 
    \begin{align*}
        \dpn{T^{**}S^*x - T^{**}x, \phi}{F} &= \dpn{T^{**}S^*x, \phi}{F} - \dpn{Tx, \phi}{F} \\
        &= \dpn{x, ST^*\phi}{E} - \dpn{x, T^*\phi}{E} \\
        |\dpn{T^{**}S^*x - Tx, \phi}{F}| &\le \eps \norm{x}_E
    \end{align*}

    As this holds for all $\phi \in B_{F^*}(0, 1)$, $\norm{T^{**}S^*x - Tx}_F \le \eps \norm{x}_E$ by \autoref{proposition:dual-norm}. Therefore $\norm{T^{**}S^* - T}_{L(E; F)} \le \eps$. 
    
    (2*) $\Rightarrow$ (1*): Let $A \subset E^*$ be compact. Using \hyperref[Mazur's Theorem]{theorem:convex-hull-complete}, assume without loss of generality that $A$ is also convex and circled. 

    Since $A$ is compact, it is norm bounded and hence equicontinuous, so the polar $U := A^\circ \in \cn_E(0)$ with respect to $\dpn{E, E^*}{E}$ is a convex and circled neighbourhood of $0$.

    The canonical projection $\pi_U: E \to E_U$ induces an adjoint map $\pi_U^*: (E_U)^* \to E^*$. For each $\phi \in (E_U)^*$ with $\norm{\phi}_{(E_U)^*} \le 1$, $\pi_U^*\phi = \phi \circ \pi_U \in U^\circ$. As $A$ is already compact, convex, and circled, the \hyperref[Bipolar Theorem]{theorem:bipolar} implies that $U^{\circ} = A^{\circ\circ} = A$ and $\phi \in A$. Hence $\pi_U^* \in L((E_U)^*; (E^*)_A)$. On the other hand, for any $\phi \in A$, $U \subset \phi^{-1}(B_K(0, 1))$. As such, $\phi$ factors through $E_U$ as follows: 
    \[
    \xymatrix{
    E \ar@{->}[r]^{\pi_U} \ar@{->}[rd]_{\phi} & E_U \ar@{->}[d]^{\widehat \phi} \\
    & K
    }
    \]

    where $\normn{\wh \phi}_{(E_U)^*} \le 1$. Thus $\pi_U^*$ is an isomorphism between $(E_U)^*$ and $(E^*)_A$. 
    
    Identify $(E_U)^*$ with $(E^*)_A$, then the inclusion $\iota_A: (E^*)_A \to E^*$ corresponds exactly to the adjoint of $\pi_U: E \to E_U$. Since $A$ is compact, $\iota_A: (E^*)_A \to E^*$ is compact, so \hyperref[Schauder's Theorem]{theorem:compact-adjoint} implies that $\pi_U: E \to \wh E_U$ is compact as well. 

    Let $\eps > 0$, then by assumption applied to $\pi_U \in \mathcal{K}(E; \wh E_U)$, there exists $T \in E^* \otimes \wh E_U$ such that $\norm{T - \pi_U}_{L(E; \wh E_U)} \le \eps$. In which case, $T^* \in (E_U)^{**} \otimes E^{*} = (E^*)_A^* \otimes E^*$ with $\norm{T^* - \iota_A}_{L((E^*)_A; E^*)} \le \eps$ as well. 
    
    Finally, since $A$ is compact, \hyperref[Goldstine's Theorem]{theorem:goldstine-weak} and the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli} allow assuming without loss of generality that $T^*$ takes the form of an element of $E_U \otimes E^*$ on $(E^*)_A$. In which case, $T^*$ indeed corresponds to an element of $E^{**} \otimes E^*$ such that $\norm{T^*\phi - \phi}_{E^*} \le \eps$ for all $\phi \in A$. 
\end{proof}

\begin{corollary}
\label{corollary:approximation-property-dual}
    Let $E$ be a Banach space over $K \in \RC$. If $E^*$ has the approximation property, then so does $E$. 
\end{corollary}
\begin{proof}
    By (3) of \autoref{theorem:approximation-property-dual}, for any Banach space $F$, the canonical map from $F^{*} \wh \otimes_\pi E^*$ to $L(F; E^*)$ is injective. Since $L(F; E^*)$ is canonically isomorphic to $L(E; F^*)$, the canonical map from $F^* \wh \otimes_\pi E^*$ to $L(E; F^*)$ is then injective. 

    Now, let $F := E^*$, then the above yields an injection from $E^{**} \wh \otimes_\pi E^*$ to $L(E; E^{**})$. Let $T \in E \wh \otimes_\pi E^*$. By \autoref{theorem:metrisable-tensor-product}, there exists $\seq{x_n} \subset E$ and $\seq{\phi_n} \subset E^*$ such that $\sum_{n \in \natp}\norm{x_n}_{E}\norm{\phi_n}_{E^*} < \infty$ and $T = \sum_{n =1}^\infty x_n \otimes \phi_n$. As an operator, for each $x \in E$, 
    \[
    Tx = \sum_{n = 1}^\infty x_n \dpn{x, \phi_n}{E} \in E
    \]

    Therefore the restriction of the canonical map $E^{**} \wh \otimes_\pi E^* \to L(E; E^{**})$ to $E \wh \otimes_\pi E^*$ yields an injection into $L(E; E)$. By (4) of \autoref{theorem:approximation-property-dual}, $E$ has the approximation property. 
\end{proof}

\begin{corollary}
\label{corollary:approximation-property-nuclear}
    Let $E$ and $F$ be Banach spaces over $K \in \RC$. If $E^*$ or $F$ has the approximation property, then the canonical map
    \[
    E^* \otimes_\pi F \to N(E; F) \quad \braks{\sum_{j = 1}^n \phi_j \otimes y_j}(x) = \sum_{j = 1}^n y_j \dpn{x, \phi_j}{E}
    \]

    extends to an isometric isomorphism. 
\end{corollary}
\begin{proof}
    If $F$ has the approximation property, then the canonical map $E^* \wh \otimes_\pi F \to N(E; F)$ is injective by (3) of \autoref{theorem:approximation-property-dual}. 

    If $E^*$ has the approximation property, then by (3) of \autoref{theorem:approximation-property-dual}, the canonical map
    \[
    F^{**} \wh \otimes_\pi E^{*} \to N(F^*; E^*) \iso N(E; F^{**})
    \]
    
    is injective. Restricting to $F \wh \otimes_\pi E^*$ yields an injection into $N(E; F)$. 
\end{proof}

\begin{corollary}[Existence of Continuous Trace]
\label{corollary:trace-existence-approx}
    Let $E$ a Banach space over $K \in \RC$ with the approximation property, then there exists a unique $\tr \in N(E; E)^*$ such that for each $\phi \in E^*$ and $x \in E$, $\tr(\phi \otimes y) = \dpn{y, \phi}{E}$.
\end{corollary}
\begin{proof}
    By (U) of the \hyperref[projective tensor product]{definition:projective-tensor-product} and the isomorphism $E^* \wh \otimes_\pi E \iso N(E; E)$ from \autoref{corollary:approximation-property-nuclear}. 
\end{proof}