14.9 The Approximation Property

Proof. (1) $\Rightarrow$ (2): Let $T \in L_{c}(E; E)$ and $A \subset E$ be precompact, then $T(A)$ is also precompact by Proposition 6.4.2. 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 $x \in A$.$\square$

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 $x \in A$,

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 $Rx - x \in 3V$.$\square$

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 Lemma 13.3.7, 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, Mazur’s Theorem implies that $\ol{\conv}(\seq{x_n})$ is compact as well. Thus for each $T \in L(E; F)$,

In particular,

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:

The Hahn-Banach Theorem then yields an extension $\Phi$ of $\wh \phi$ as shown below:

By Theorem 16.5.5, 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 $T \in L(E; F)$,

$\square$

Proof, [Theorem III.9.5, SW99] and [Proposition 4.6, Rya02]. (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 Lemma 13.3.7, 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 $E_{B}$, Goldstine’s Theorem and the Arzelà-Ascoli Theorem 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,

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 Theorem 13.11.4, 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 Proposition 13.11.2,

Since $\sum_{n \in \natp}\norm{\phi_n}_{F^*}< \infty$, assume without loss of generality that $\bracsn{y_k}_{1}^{m} \subset F$ with Goldstine’s Theorem. This allows rewriting

so

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 Hahn-Banach Theorem, 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 Lemma 14.9.4, there exists a null sequence $\seq{x_n}\subset E$ and $\seq{\psi_n}\in l^{1}(\natp; E^{*})$ such that for each $T \in L(E; E)$,

In particular, for any $x \in E$ and $\eta \in E^{*}$,

By the Hahn-Banach Theorem, $\sum_{n = 1}^{\infty} x_{n} \dpn{x, \psi_n}{E}= 0$. 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 Schauder’s Theorem, 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 Gantmacher’s Theorem, $T^{**}(E^{**}) \subset F$. Thus $T^{**}S^{*} \in E^{***}\otimes F \subset L(E, F)$. For any $x \in E$ and $\phi \in B_{F^*}(0, 1)$,

As this holds for all $\phi \in B_{F^*}(0, 1)$, $\norm{T^{**}S^*x - Tx}_{F} \le \eps \norm{x}_{E}$ by Proposition 14.1.7. Therefore $\norm{T^{**}S^* - T}_{L(E; F)}\le \eps$.

(2*) $\Rightarrow$ (1*): Let $A \subset E^{*}$ be compact. Using Mazur’s Theorem, 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 Bipolar Theorem 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:

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 Schauder’s Theorem 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, Goldstine’s Theorem and the Arzelà-Ascoli Theorem 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 $\phi \in A$.$\square$

Proof. By (3) of Theorem 14.9.5, 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 Theorem 13.11.4, 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 $x \in E$,

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 Theorem 14.9.5, $E$ has the approximation property.$\square$

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 Theorem 14.9.5.

If $E^{*}$ has the approximation property, then by (3) of Theorem 14.9.5, the canonical map

is injective. Restricting to $F \wh \otimes_{\pi} E^{*}$ yields an injection into $N(E; F)$.$\square$

Proof. By (U) of the projective tensor product and the isomorphism $E^{*} \wh \otimes_{\pi} E \iso N(E; E)$ from Corollary 14.9.7.$\square$