Duality in Banach Spaces

Let $\phi$ be the embedding of $X$ into $X^{* *}$ described in Section 4.5. Let $\tau$ be the weak topology of $X$, and let $\sigma$ be the weak*-topology of $X^{* *}$-the one induced by $X^{*}$.

Which of the spaces $c_{0}, \ell^{1}, \ell^{p}, \ell^{\infty }$ are reflexive? Prove that every finitedimensional normed space is reflexive. Prove that $C$, the supremum-normed space of all complex continuous functions, on the unit interval, is not reflexive.

Prove that a subset $E$ of $\mathscr {B}(X, Y)$ is equicontinuous if and only if there exists $M < \infty$ such that $\left\| \Lambda \right\| \leq M$ for every $\Lambda \in E$.

Recall that $X^{*} = \mathscr {B}(X, \mathscr {C})$, if $\mathscr {C}$ is the scalar field. Hence $\Lambda^{*} \in \mathscr {B}\left(\mathbb {C}, X^{*}\right)$ for every $\Lambda \in X^{*}$. Identify the range of $\Lambda^{*}$.

Prove that $T \in \mathscr {B}(X, Y)$ is an isometry of $X$ onto $Y$ if and only if $T^{*}$ is an isometry of $Y^{*}$ onto $X^{*}$.

Let $\sigma$ and $\tau$ be the weak*-topologies of $X^{*}$ and $Y^{*}$, respectively, and prove that $S$ is a continuous linear mapping of $\left(Y^{*}, \tau \right)$ into $\left(X^{*}, \sigma \right)$ if and only if $S = T^{*}$ for some $T \in \mathscr {B}(X, Y)$.

Let $L^{1}$ be the usual space of integrable functions on the closed unit interval $J$, relative to Lebesgue measure. Suppose $T \in \mathscr {B}\left(L^{1}, Y\right)$, so that $T^{*} \in \mathscr {B}\left(Y^{*}, L^{\infty }\right)$. Suppose $\mathscr {R}\left(T^{*}\right)$ contains every continuous function on $J$. What can you deduce about $T$?

Prove that $(S T)^{*} = T^{*} S^{*}$. Supply the hypotheses under which this makes sense.

Suppose $S \in \mathscr {B}(X), T \in \mathscr {B}(X)$.

Assume $T \in \mathscr {B}(X)$ is compact, and assume either that $\operatorname {dim} X = \infty$ or that the scalar field is $\mathbb {C}$. Prove that $\sigma (T)$ is not empty. However, $\sigma (T)$ may be empty if $\operatorname {dim} X < \infty$ and the scalar field is $\mathbb {R}$.

Suppose $\operatorname {dim} X < \infty$ and show that the equality $\beta^{*} = \beta$ of Theorem 4.25 reduces to the statement that the row rank of a square matrix is equal to its column rank.

Suppose $T \in \mathscr {B}(X, Y)$ and $\mathscr {R}(T)$ is closed in $Y$. Prove that

This generalizes the assertions $\alpha = \beta^{*}$ and $\alpha^{*} = \beta$ of Theorem 4.25.

Define a shift operator $S$ and a multiplication operator $M$ on $\ell^{2}$ by

Put $T = M S$. Show that $T$ is a compact operator which has no eigenvalue and whose spectrum consists of exactly one point. Compute $\left\| T^{n}\right\|$, for $n = 1,2,3, \ldots$, and compute $\lim_{n \rightarrow \infty }\left\| T^{n}\right\|^{1 / n}$.

Suppose $\mu$ is a finite (or $\sigma$-finite) positive measure on a measure space $\Omega , \mu \times \mu$ is the corresponding product measure on $\Omega \times \Omega$, and $K \in L^{2}(\mu \times \mu )$. Define

Define

and define $T \in \mathscr {B}\left(L^{2}(0,1)\right)$ by

If $L^{2} = L^{2}(0, \infty )$ relative to Lebesgue measure, and if

prove that $T \in \mathscr {B}\left(L^{2}\right)$ and that $T$ is not compact. (The fact that $\left\| T\right\| \leq 2$ is a special case of Hardy's inequality. See p. 72 of [23]).

Prove the following statements:

Suppose $Y$ is a closed subspace of $X$, and $x_{0}^{*} \in X^{*}$. Put

In other words, $\mu$ is the norm of the restriction of $x_{0}^{*}$ to $Y$, and $\delta$ is the distance from $x_{0}^{*}$ to the annihilator of $Y$. Prove that $\mu = \delta$. Prove also that $\delta = \left\| x^{*}-x_{0}^{*}\right\|$ for at least one $x^{*} \in Y^{\perp }$.

Extend Sections 4.6 to 4.9 to locally convex spaces. (The word "isometric" must of course be deleted from the statement of Theorem 4.9).

Let $B$ and $B^{*}$ be the closed unit balls in $X$ and $X^{*}$, respectively. The following is a converse of the Banach-Alaoglu theorem: If $E$ is a convex set in $X^{*}$ such that $E \cap \left(r B^{*}\right)$ is weak*-compact for every $r > 0$, then $E$ is weak*-closed. (Corollary: A subspace of $X^{*}$ is weak*-closed if and only if its intersection with $B^{*}$ is weak*-compact).

Complete the following outline of the proof.

(i) $E$ is norm-closed.

(ii) Associated to each $F \subset X$ its polar

The intersection of all sets $P(F)$, as $F$ ranges over the collection of all finite subsets of $r^{-1} B$, is exactly $r B^{*}$.

(iii) The theorem is a consequence of the following proposition: If, in addition to the stated hypotheses, $E \cap B^{*} = \varnothing$, then there exists $x \in X$ such that $\mathrm{Re}\left( \left\langle x, x^{*} \right\rangle \right) \geq 1$ for every $x^{*} \in E$.

(iv) Proof of the proposition: Put $F_{0} = \left\{ 0\right\}$. Assume finite sets $F_{0}, \ldots , F_{k-1}$ have been chosen so that $i F_{i} \subset B$ and so that

Note that (1) is true for $k = 1$. Put

If $P(F) \cap Q \neq \varnothing$ for every finite set $F \subset k^{-1} B$, the weak*-compactness of $Q$, together with (ii), implies that $\left(k B^{*}\right) \cap Q \neq \varnothing$, which contradicts (1). Hence there is a finite set $F_{k} \subset k^{-1} B$ such that (1) holds with $k+1$ in place of $k$. The construction can thus proceed. It yields

Arrange the members of $\bigcup F_{k}$ in a sequence $\left\{ x_{n}\right\}$. Then $\left\| x_{n}\right\| \rightarrow 0$. Define $T: X^{*} \rightarrow c_{0}$ by

Then $T(E)$ is a convex subset of $c_{0}$. By (2),

for every $x^{*} \in E$. Hence there is a scalar sequence $\left\{ \alpha_{n}\right\}$, with $\sum \left|\alpha_{n}\right| < \infty$, such that

for every $x^{*} \in E$. To complete the proof, put $x = \sum \alpha_{n} x_{n}$.

Suppose $T \in \mathscr {B}(X), T$ is compact, $\lambda \neq 0$, and $S = T-\lambda I$.

Suppose $\left\{ x_{n}\right\}$ is a sequence in a Banach space $X$, and

Prove that the series $\sum x_{n}$ converges to some $x \in X$. Explicitly, prove that

Prove also that $\left\| x\right\| \leq M$. (These facts were used in the proof of Theorem 4.13).

Let $c$ be the space of all complex sequences

for which $x_{\infty } = \lim x_{n}$ exists (in $\mathbb {C}$). Put $\left\| x\right\| = \sup \left|x_{n}\right|$. Let $c_{0}$ be the subspace of $c$ that consists of all $x$ with $x_{\infty } = 0$.

If $T \in \mathscr {B}(X, Y)$ and $\mathscr {R}\left(T^{*}\right) = \mathscr {N}(T)^{\perp }$, prove that $\mathscr {R}(T)$ is closed.

Assume $T \in \mathscr {B}(X, Y)$ and $T(X) = Y$. Show that there exists $\delta > 0$ such that $S(X) = Y$ for all $S \in \mathscr {B}(X, Y)$ with $\left\| S-T\right\| < \delta$.

Suppose $T \in \mathscr {B}(X)$. Prove that $\lambda \in \sigma (T)$ if and only if there is a sequence $\left\{ x_{n}\right\}$ in $X,\left\| x_{n}\right\| = 1$, for which