Added characterisation of the approximation property.
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Bokuan Li
11
refs.bib
11
refs.bib
@@ -257,4 +257,15 @@
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address = {Upper Saddle River, NJ},
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year = {2000},
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isbn = {0-13-181629-2}
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}
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@book{RyanTensor,
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author = {Ryan, Raymond A.},
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title = {Introduction to Tensor Products of Banach Spaces},
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series = {Springer Monographs in Mathematics},
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publisher = {Springer},
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address = {London},
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year = {2002},
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isbn = {978-1-85233-437-6},
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doi = {10.1007/978-1-4471-3903-4}
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}
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@@ -75,8 +75,8 @@
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\angles{\sum_{k = 1}^n x_k \otimes y_k, T}_{E \otimes_\pi F} = \sum_{k = 1}^n \dpn{x_k, Ty_k}{E}
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\]
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\end{enumerate}
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\end{proposition}
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% Proof omitted because the hardest part of this result is writing it down.
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@@ -95,9 +95,9 @@
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is an isometric isomorphism.
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\end{theorem}
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\begin{proof}
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By \hyperref[Hölder's Inequality]{proposition:lp-direct-sum-gymnastics}, for each $y \in [l^1(I); X_i^*]$, $\norm{\phi_y}_{[c_0(I); X_i]^*} \le \norm{y}_{[l^1(I); X_i]}$.
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By \hyperref[Hölder's Inequality]{proposition:lp-direct-sum-gymnastics}, for each $y \in [l^1(I); X_i^*]$, $\norm{\phi_y}_{[c_0(I); X_i]^*} \le \norm{y}_{[l^1(I); X_i^*]}$.
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Let $\phi \in [c_0(I); X_i]^*$, then there exists $y \in [l^\infty(I); X_i^*]$ such that for each $i \in I$ and $x \in X_i$, $\dpn{x_i \cdot \one_{\bracs{i}}, \phi}{[l^p(I); X_i]} = \dpn{x_i, y_i}{X_i}$.
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Let $\phi \in [c_0(I); X_i]^*$, then there exists $y \in [l^\infty(I); X_i^*]$ such that for each $i \in I$ and $x_i \in X_i$, $\dpn{x_i \cdot \one_{\bracs{i}}, \phi}{[c_0(I); X_i]} = \dpn{x_i, y_i}{X_i}$.
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Let $J \subset I$ be finite and $\alpha \in (0, 1)$, then there exists $x \in [c_0(I); X_i]$ such that
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\begin{enumerate}
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@@ -108,10 +108,10 @@
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Thus
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\[
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\alpha\sum_{j \in J}\norm{y_j}_{X_j^*} \le \sum_{j \in J}\dpn{x_j, y_j}{X_j} = \dpn{x, y}_{[c_0(I); X_i]} \le \norm{\phi}_{[c_0(I); X_i]^*}
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\alpha\sum_{j \in J}\norm{y_j}_{X_j^*} \le \sum_{j \in J}\dpn{x_j, y_j}{X_j} = \dpn{x, \phi}{[c_0(I); X_i]} \le \norm{\phi}_{[c_0(I); X_i]^*}
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\]
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As the above holds for all $\alpha \in (0, 1)$ and $J \subset I$ finite, $y \in [l^1(I); X_i]$ with $\norm{y}_{[l^1(I); X_i^*]} \le \norm{\phi_y}_{[c_0(I); X_i]^*}$. By the \hyperref[Dominated Convergence Theorem]{theorem:dct}, $\dpn{x, \phi_y}{[c_0(I); X_i]} = \dpn{x, \phi}{[c_0(I); X_i]}$ for all $x \in [c_0(I); X_i]$. Hence the map is an isometric isomorphism.
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As the above holds for all $\alpha \in (0, 1)$ and $J \subset I$ finite, $y \in [l^1(I); X_i^*]$ with $\norm{y}_{[l^1(I); X_i^*]} \le \norm{\phi}_{[c_0(I); X_i]^*}$. By the \hyperref[Dominated Convergence Theorem]{theorem:dct}, $\dpn{x, \phi_y}{[c_0(I); X_i]} = \dpn{x, \phi}{[c_0(I); X_i]}$ for all $x \in [c_0(I); X_i]$. Hence the map is an isometric isomorphism.
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\end{proof}
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@@ -130,7 +130,7 @@
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is an isometric isomorphism.
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\end{theorem}
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\begin{proof}
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Let $\phi \in [l^p(I); X_i]^*$, then there exists $y \in [l^\infty(I); X_i^*]$ such that for each $i \in I$ and $x \in X_i$, $\dpn{x_i \cdot \one_{\bracs{i}}, \phi}{[l^p(I); X_i]} = \dpn{x_i, y_i}{X_i}$.
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Let $\phi \in [l^p(I); X_i]^*$, then there exists $y \in [l^\infty(I); X_i^*]$ such that for each $i \in I$ and $x_i \in X_i$, $\dpn{x_i \cdot \one_{\bracs{i}}, \phi}{[l^p(I); X_i]} = \dpn{x_i, y_i}{X_i}$.
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Since the $q = \infty$ case has been ruled out, assume that $q \in (1, \infty)$. For each $\alpha \in (0, 1)$, there exists $x \in [l^\infty(I); X_i]$ with $\norm{x_i}_{X_i} \le 1$ and $\dpn{x_i, y_i}{X_i} \ge \alpha \norm{y_i}_{X_i^*}$. For each $J \subset I$ finite and $i \in I$, let $F_J(i) = \one_{J}(i) \cdot \norm{y_i}_{X_i^*}^{q - 1}$, then by \autoref{lemma:holder-conjugate-gymnastics},
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\[
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@@ -41,5 +41,159 @@
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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.
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\end{corollary}
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\begin{lemma}
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\label{lemma:compact-operator-topology-banach-dual}
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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:
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\begin{enumerate}
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\item $\limv{n}x_n = 0$.
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\item $\sum_{n \in \natp}\norm{\psi_n}_{F^*} < \infty$.
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\item For each $T \in L(E; F)$, $\dpn{T, \phi}{L_c(E; F)} = \sum_{n = 1}^\infty \dpn{Tx_n, \psi_n}{F}$.
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\end{enumerate}
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\end{lemma}
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\begin{proof}
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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 $T \in L(E; F)$.
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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 $T \in L(E; F)$,
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\[
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T(A) \subset T(\ol{\conv}(\seq{x_n})) = \ol{\conv}(T(\seq{x_n}))
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\]
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In particular,
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\[
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|\dpn{T, \phi}{L_c(E; F)}| \le \sup_{x \in A}\norm{Tx}_F \le \sup_{n \in \natp}\norm{Tx_n}_F
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\]
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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:
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\[
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\xymatrix{
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L_c(E; F) \ar@{->}[rd]_{\phi} \ar@{->}[r] & L \ar@{->}[d]^{\widehat \phi} \\
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& K
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}
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\]
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The \hyperref[Hahn-Banach Theorem]{theorem:hahn-banach} then yields an extension $\Phi$ of $\wh \phi$ as shown below:
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\[
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\xymatrix{
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L_c(E; F) \ar@{->}[rd]_{\phi} \ar@{->}[r] & L \ar@{->}[d]^{\widehat \phi} \ar@{->}[r] & c_0(\mathbb{N}^+; F) \ar@{->}[ld]^{\Phi} \\
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& K &
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}
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\]
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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 $T \in L(E; F)$,
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\begin{align*}
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\dpn{T, \phi}{L_c(E; F)} &= \dpn{\seq{Tx_n}, \wh \phi}{L} = \dpn{\seq{Tx_n}, \Phi}{c_0(\natp; F)} \\
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&= \sum_{n = 1}^\infty \dpn{Tx_n, \psi_n}{F}
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\end{align*}
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\end{proof}
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\begin{remark}
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\label{remark:compact-operator-topology-banach-dual}
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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.
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\end{remark}
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\begin{theorem}
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\label{theorem:approximation-property-dual}
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Let $E$ be a Banach space over $K \in \RC$, then the following are equivalent:
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\begin{enumerate}
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\item $E$ has the approximation property.
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\item For any Banach space $F$, the closure of $F^* \otimes E$ in $L(F; E)$ is $\mathcal{K}(F; E)$.
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\item For any Banach space $F$, the canonical map $F^* \wh \otimes_\pi E \to L(F; E)$ is injective.
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\end{enumerate}
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and the following are equivalent:
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\begin{enumerate}[label=(\arabic**)]
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\item $E^*$ has the approximation property.
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\item For any Banach space $F$, the closure of $E^* \otimes F$ in $L(E; F)$ is $\mathcal{K}(E; F)$.
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\end{enumerate}
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\end{theorem}
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\begin{proof}[Proof, {{\cite[Theorem III.9.5]{SchaeferWolff}}} and {{\cite[Proposition 4.6]{RyanTensor}}}. ]
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(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 $\norm{ST - T}_{L(F; E)} \le \eps$.
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(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 $E_B$.
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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 $x \in E_B$.
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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,
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\begin{align*}
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\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 \\
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&+ \sum_{j = 1}^n \norm{x_j}_E|\dpn{x, \phi_j - \psi_j}{E_B}| \\
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&\le \eps\norm{x}_{E_B} + \eps \le \eps\braks{1 + \sup_{x \in A}\norm{x}_{E_B}}
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\end{align*}
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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.
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(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 $\sum_{n \in \natp}\norm{\phi_n}_{F^*} < \infty$.
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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},
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\begin{align*}
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\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^*} \\
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&= \sum_{k = 1}^m \sum_{n = 1}^\infty \dpn{\phi_n, y_k}{F^*} \dpn{x_n, \psi_k}{E}
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\end{align*}
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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
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\[
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\dpn{T, R}{F^* \wh \otimes_\pi E} = \sum_{k = 1}^m \dpn{Ty_k, \psi_k}{E} = 0
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\]
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so
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\[
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|\dpn{T, S}{F^* \wh \otimes_\pi E}| \le |\dpn{T, R}{F^* \wh \otimes_\pi E}| + \eps \sum_{n \in \natp}\norm{\phi_n}_{F^*} = \eps \sum_{n \in \natp}\norm{\phi_n}_{F^*}
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\]
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As the above holds for all $\eps > 0$, $\dpn{T, S}{F^* \wh \otimes_\pi E} = 0$. Therefore $T = 0$ as an element of $F^* \wh \otimes_\pi E$.
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$\neg$ (1) $\Rightarrow$ $\neg$ (3): 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 $T \in E^* \otimes E$.
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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 $T \in L(E; E)$,
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\[
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\dpn{T, \phi}{L_c(E; E)} = \sum_{n = 1}^\infty \dpn{Tx_n, \psi_n}{E}
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\]
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In particular, for any $x \in E$ and $\eta \in E^*$,
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\begin{align*}
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0 &= \dpn{\eta \otimes x, \phi}{L_c(E; E)} = \sum_{n = 1}^\infty \dpn{x_n, \eta}{E} \dpn{x, \psi_n}{E} \\
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&= \angles{\sum_{n = 1}^\infty x_n\dpn{x, \psi_n}{E}, \eta}_E
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\end{align*}
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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 $x \in E$.
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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 $x \in E$.
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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.
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(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.
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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 $\phi \in T^*(B_{F^*}(0, 1))$.
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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 $\phi \in B_{F^*}(0, 1)$,
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\begin{align*}
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\dpn{T^{**}S^*x - T^{**}x, \phi}{F} &= \dpn{T^{**}S^*x, \phi}{F} - \dpn{Tx, \phi}{F} \\
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&= \dpn{x, ST^*\phi}{E} - \dpn{x, T^*\phi}{E} \\
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|\dpn{T^{**}S^*x - Tx, \phi}{F}| &\le \eps \norm{x}_E
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\end{align*}
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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 $\norm{T^{**}S^* - T}_{L(E; F)} \le \eps$.
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(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.
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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 $0$.
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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:
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\[
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\xymatrix{
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E \ar@{->}[r]^{\pi_U} \ar@{->}[rd]_{\phi} & E_U \ar@{->}[d]^{\widehat \phi} \\
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& K
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}
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\]
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where $\normn{\wh \phi}_{(E_U)^*} \le 1$. Thus $\pi_U^*$ is an isomorphism between $(E_U)^*$ and $(E^*)_A$.
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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.
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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.
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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 $\phi \in A$.
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\end{proof}
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Reference in New Issue
Block a user