Math Problems Vault

Problem VI.5.1

(a)(b)(c)

Prove Proposition 5.2.

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(a)

If either $\mathscr {X}$ or $\mathscr {Y}$ is reflexive, then every operator in $\mathcal{B}(\mathscr {X}, \mathscr {Y})$ is weakly compact.

(b)

If $T: \mathscr {X} \to \mathscr {Y}$ is weakly compact and $A \in \mathcal{B}(\mathscr {Y}, \mathscr {Z})$ , then $AT$ is weakly compact.

(c)

If $T: \mathscr {X} \to \mathscr {Y}$ is weakly compact and $B \in \mathcal{B}(\mathscr {Z}, \mathscr {X})$ , then $TB$ is weakly compact.

Problem VI.5.2

If $\mathscr {R}$ and $\mathscr {X}$ are Banach spaces and $\Phi : \mathscr {R} \to \mathscr {X}$ is an isometry, give an elementary proof that $\Phi^*$ is surjective.

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Problem VI.5.3

(a)(b)

Let $\mathscr {X}$ be a Banach space and recall the definition of a weakly Cauchy sequence (V.4.4).

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(a)

Show that every bounded sequence in $c_0$ has a weakly Cauchy subsequence, but not every weakly Cauchy sequence in $c_0$ converges.

(b)

Show that if $T \in \mathcal{B}(c_0)$ and $T$ is weakly compact, then $T$ is compact.

Problem VI.5.4

Say that a Banach space $\mathscr {X}$ is weakly compactly generated (WCG) if there is a weakly compact subset $K$ of $\mathscr {X}$ such that $\mathscr {X}$ is the closed linear span of $K$ . Prove (Davis, Figiel, Johnson, and Pelczynski, [1974]) that $\mathscr {X}$ is WCG if and only if there is a reflexive space and an injective bounded operator $T: \mathscr {R} \to \mathscr {X}$ such that $\mathrm{ran}\ T$ is dense.

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Problem VI.5.5

If $(X, \Omega , \mu )$ is a finite measure space, $k \in L^\infty (X \times X, \Omega \times \Omega , \mu \times \mu )$ , and $K: L^1(\mu ) \to L^1(\mu )$ is defined by $(Kf)(x) = \int k(x, y) f(y)\, d\mu (y)$ , show that $K$ is weakly compact and $K^2$ is compact.

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Problem VI.5.6

(a)(b)(c)

Let $\mathscr {Y}$ be a weakly sequentially complete Banach space. That is, if $\left\{ y_n\right\}$ is a sequence in $\mathscr {Y}$ such that $\left\{ \langle y_n, y^* \rangle \right\}$ is a Cauchy sequence in $\mathbb {F}$ for every $y^*$ in $\mathscr {Y}^*$ , then there is a $y$ in $\mathscr {Y}$ such that $y_n \to y$ weakly.

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(a)

If $T \in \mathcal{B}(\mathscr {X}, \mathscr {Y})$ and $x^{**} \in \mathscr {X}^{**}$ such that $x^{**}$ is the $\sigma (\mathscr {X}^{**}, \mathscr {X}^*)$ limit of a sequence $\left\{ x_n^{**}\right\}$ from $\mathscr {X}^{**}$ such that $T^{**}(x_n^{**}) \in \mathscr {Y}$ for every $n$ , show that $T^{**}(x^{**}) \in \mathscr {Y}$ . Let $X$ be a compact space and put $\mathscr {F} =$ all subsets of $X$ that are the union of a countable number of compact $G_\delta$ sets. Let $\mathscr {L} =$ the linear span of $\left\{ \chi_F: F \in \mathscr {F}\right\}$ considered as a subset of $M(X)^* = C(X)^{**}$ .

(b)

Show that if $T \in \mathcal{B}(C(X), \mathscr {Y})$ , then $T^{**}(\mathscr {L}) \subseteq \mathscr {Y}$ .

(c)

(Grothendieck [1953].) If $T \in \mathcal{B}(C(X), \mathscr {Y})$ , then $T$ is weakly compact.