Weak Topologies

Show that wk is the smallest topology on $\mathcal{X}$ such that each $x^*$ in $\mathcal{X}^*$ is continuous.

Show that wk* is the smallest topology on $\mathcal{X}^*$ such that for each $x$ in $\mathcal{X}$, $x^* \mapsto \left\langle x, x^* \right\rangle$ is continuous.

Prove Theorem 1.3.

Let $\mathcal{X}$ be a complex LCS and let $\mathcal{X}^*_{\mathbb {R}}$ denote the collection of all continuous real linear functionals on $\mathcal{X}$. Use the elements of $\mathcal{X}^*_{\mathbb {R}}$ to define seminorms on $\mathcal{X}$ and let $\sigma (\mathcal{X}, \mathcal{X}^*_{\mathbb {R}})$ be the corresponding topology. Show that $\sigma (\mathcal{X}, \mathcal{X}^*) = \sigma (\mathcal{X}, \mathcal{X}^*_{\mathbb {R}})$.

Prove the remainder of Proposition 1.7.

If $A \subseteq \mathcal{X}$, show that $A$ is weakly bounded if and only if $A^\circ$ is absorbing in $\mathcal{X}^*$.

Let $\mathcal{X}$ be a normed space and let $\left\{ x_n\right\}$ be a sequence in $\mathcal{X}$ such that $x_n \to x$ weakly. Show that there is a sequence $\left\{ y_n\right\}$ such that $y_n \in \operatorname {co}\left\{ x_1, x_2, \dots , x_n\right\}$ and $\left\| y_n - x\right\| \to 0$.

If $\mathcal{H}$ is a Hilbert space and $\left\{ h_n\right\}$ is a sequence in $\mathcal{H}$ such that $h_n \to h$ weakly and $\left\| h_n\right\| \to \left\| h\right\|$, then $\left\| h_n - h\right\| \to 0$. (The same type of result is true for $L^p$-spaces if $1 < p < \infty$. See W.P. Novinger [1972].)

If $\mathcal{X}$ is a normed space show that the norm on $\mathcal{X}$ is lower semicontinuous for the weak topology and the norm of $\mathcal{X}^*$ is lower semicontinuous for the weak-star topology.

Suppose $\mathcal{X}$ is an infinite-dimensional normed space. If $S = \left\{ x \in \mathcal{X} : \left\| x\right\| = 1\right\}$, then the weak closure of $S$ is $\left\{ x : \left\| x\right\| \leq 1\right\}$.

In relation to Claim 2.4, show that if $\mathcal{Z} \leq \mathcal{X}$, $\dim \mathcal{Z} < \infty$, and $\mathcal{M} \leq \mathcal{X}$, then $\mathcal{Z} + \mathcal{M}$ is closed.

Show that if $\mathcal{M} \leq \mathcal{X}$ and $\mathcal{M}$ is topologically complemented in $\mathcal{X}$, then $\mathcal{M}^\perp$ is topologically complemented in $\mathcal{X}^*$ and that its complement is weak-star and linearly homeomorphic to $\mathcal{X}^*/\mathcal{M}^\perp$.

Show that the functional $f$ occurring in the proof of Alaoglu's Theorem is linear.

Let $\mathcal{X}$ be a LCS and let $V$ be an open neighborhood of $0$. Show that $V^\circ$ is weak-star compact in $\mathcal{X}^*$.

If $\mathcal{X}$ is a Banach space, show that there is a compact space $X$ such that $\mathcal{X}$ is isometrically isomorphic to a closed subspace of $C(X)$.

Show that if $\mathcal{X}$ is reflexive and $\mathcal{M} \leq \mathcal{X}$, then $\mathcal{X}/\mathcal{M}$ is reflexive.

If $\mathcal{X}$ is a Banach space, $\mathcal{M} \leq \mathcal{X}$, and both $\mathcal{M}$ and $\mathcal{X}/\mathcal{M}$ are reflexive, must $\mathcal{X}$ be reflexive?

If $(X, \Omega , \mu )$ is a $\sigma$-finite measure space, show that $L^1(X, \Omega , \mu )$ is reflexive if and only if it is finite dimensional.

Give the details of the proofs of the statements made in Example 4.5.

Verify the statement made in Example 4.8.

If $(X, \Omega , \mu )$ is a $\sigma$-finite measure space, show that $L^\infty (\mu )$ is weak-star sequentially complete but is reflexive if and only if it is finite dimensional.

Let $X$ be compact and suppose there is a norm on $C(X)$ that is given by an inner product making $C(X)$ into a Hilbert space such that for every $x$ in $X$ the functional $f \mapsto f(x)$ on $C(X)$ is continuous with respect to the Hilbert space norm. Show that $X$ is finite.

Let $B = \operatorname {ball} M[0,1]$ and for $\mu , \nu$ in $M[0,1]$ define

Show that $d$ is a metric on $M[0,1]$ that defines the weak-star topology on $B$ but not on $M[0,1]$.

Let $X$ be a compact space and let $\mathcal{U} = \left\{ (U,V) : U, V \text{ are open subsets of } X \text{ and } \operatorname {cl} U \subseteq V\right\}$. For $u = (U,V)$ in $\mathcal{U}$, let $f_u : X \to [0,1]$ be a continuous function such that $f_u \equiv 1$ on $\operatorname {cl} U$ and $f_u \equiv 0$ on $X \setminus V$. Show:

If $\mathcal{X}$ is a Banach space and $\mathcal{X}^*$ is separable, show that

If $B = \operatorname {ball} l^\infty$, show that

defines a metric on $B$ and that this metric defines the weak-star topology on $B$.

Use the type of argument used in the proof of the Principle of Uniform Boundedness to obtain a proof of Proposition 5.2 that does not need the Baire Category Theorem.

Show that Proposition 5.2 fails for $l^p$ if $1 < p < \infty$.

If $x \in X$ and $\delta_x(f) = f(x)$ for all $f$ in $C_b(X)$, show that $\left\| \delta_x\right\| = 1$.

Prove that a subset of a completely regular space is completely regular.

Fill in the details of the proof of Theorem 6.2 (Stone-Cech Compactification): if $X$ is completely regular, then there is a compact space $\beta X$ such that

If $X$ is completely regular, $\Omega$ is compact, and $f : X \to \Omega$ is continuous, show that there is a continuous map $f^\beta : \beta X \to \Omega$ such that $f^\beta | X = f$.

If $X$ is completely regular, show that $X$ is open in $\beta X$ if and only if $X$ is locally compact.

Let $\mathbb {N}$ have the discrete topology. Let $\left\{ r_n : n \in \mathbb {N} \right\}$ be an enumeration of the rational numbers in $[0,1]$. Let $S =$ the irrational numbers in $[0,1]$ and for each $s$ in $S$ let $\left\{ r_n : n \in N_s \right\}$ be a subsequence of $\left\{ r_n \right\}$ such that $s = \lim \left\{ r_n : n \in N_s \right\}$. Show:

Show that if $X$ is normal, $\tau \in \beta X$, and there is a sequence $\left\{ x_n \right\}$ in $X$ such that $x_n \to \tau$ in $\beta X$, then $\tau \in X$. If $X$ is not normal, is the result still true?

Let $X$ be the space of all ordinals less than the first uncountable ordinal and give $X$ the order topology. Show that $\beta X =$ the one point compactification of $X$. (You can find the pertinent definitions in Kelley [1955].)

If $(X, \Omega , \mu )$ is a $\sigma$-finite measure space and $1 < p < \infty$, then the set of extreme points of $\ball L^p(\mu )$ is $\left\{ f \in L^p(\mu ) : \left\| f\right\|_{p} = 1 \right\}$.

If $(X, \Omega , \mu )$ is a $\sigma$-finite measure space, the set of extreme points of $\ball L^1(\mu )$ is $\left\{ \alpha \chi_E : E \text{ is an atom of } \mu , \ \alpha \in \mathbb {F}, \text{ and } \left|\alpha \right| = \mu (E)^{-1} \right\}$.

If $(X, \Omega , \mu )$ is a $\sigma$-finite measure space, the set of extreme points of $\ball L^\infty (\mu )$ is $\left\{ f \in L^\infty (\mu ) : \left|f(x)\right| = 1 \text{ a.e. } [\mu ] \right\}$.

If $X$ is completely regular, the set of extreme points of $\ball C_b(X)$ is $\left\{ f \in C_b(X) : \left|f(x)\right| = 1 \text{ for all } x \right\}$. So $\ball C_{\mathbb {R}}[0,1]$ has only two extreme points.

Let $X$ be a totally disconnected compact space. (That is, $X$ is compact and if $x \in X$ and $U$ is a neighborhood of $x$, then there is a subset $V$ of $X$ that is both open and closed and such that $x \in V \subseteq U$. The Cantor set is an example of such a space.) Show that $\ball C(X)$ is the norm closure of the convex hull of its extreme points.

Show that $\ball l^1$ is the norm closure of the convex hull of its extreme points.

Show that if $X$ is locally compact but not compact, then $\ball C_0(X)$ has no extreme points.

If $\mathcal{X}$ is a LCS and $K_1, \dots , K_n$ are compact convex subsets of $\mathcal{X}$, then $\overline{\co }(K_1 \cup \dots \cup K_n) = \co (K_1 \cup \dots \cup K_n)$ and this convex hull is compact.

Let $K$ be convex and let $T : K \to \mathcal{Y}$ be an affine map. If $y$ is an extreme point of $T(K)$ and $x$ is an extreme point of $T^{-1}(y)$, then $x$ is an extreme point of $K$.

If $\mathcal{H}$ is a Hilbert space and either $T$ or $T^*$ is an isometry, show that $T$ is an extreme point of the closed unit ball of $\mathcal{B}(\mathcal{H})$.

Suppose that $\mathcal{A}$ is a subalgebra of $C(X)$ that separates the points of $X$ and $1 \in \mathcal{A}$. Show that if $x_1, \dots , x_n$ are distinct points in $X$ and $\alpha_1, \dots , \alpha_n \in \mathbb {F}$, there is an $f$ in $\mathcal{A}$ such that $f(x_j) = \alpha_j$ for $1 \leq j \leq n$.

Give the details of the proof of Corollary 8.3: if $X$ is locally compact and $\mathcal{A}$ is a closed subalgebra of $C_0(X)$ such that

If $X$ is compact, show that for each $x$ in $X$, $\delta_x$ is an extreme point of $P(X)$ and $\alpha \delta_x$, $\left|\alpha \right| = 1$, is an extreme point of $\ball M(X)$.

Let $X$ be compact and let $\mathcal{A}$ be a closed subalgebra of $C(X)$ such that $1 \in \mathcal{A}$ and $\mathcal{A}$ is closed under conjugation. Define an equivalence relation $\sim$ on $X$ by declaring $x \sim y$ if and only if $f(x) = f(y)$ for all $f$ in $\mathcal{A}$. Let $X / \sim$ be the corresponding quotient space and let $\pi : X \to X / \sim$ be the natural map. Give $X / \sim$ the quotient topology.

Let $X$ be completely regular and topologize $C(X)$ as in Example IV.1.5. If $\mathcal{A}$ is a closed subalgebra of $C(X)$ such that $1 \in \mathcal{A}$, $\mathcal{A}$ separates the points of $X$, and $\bar{f} \in \mathcal{A}$ whenever $f \in \mathcal{A}$, then $\mathcal{A} = C(X)$.

Let $X, Y$ be compact spaces and show that if $f \in C(X \times Y)$ and $\varepsilon > 0$, then there are functions $g_1, \dots , g_n$ in $C(X)$ and $h_1, \dots , h_n$ in $C(Y)$ such that $\left|f(x,y) - \sum_{k=1}^n g_k(x) h_k(y)\right| < \varepsilon$ for all $(x,y)$ in $X \times Y$.

Let $\mathcal{A}$ be the uniformly closed subalgebra of $C_b(\mathbb {R})$ generated by $\sin x$ and $\cos x$. Show that $\mathcal{A} = \left\{ f \in C_b(\mathbb {R}) : f(t) = f(t + 2\pi ) \text{ for all } t \text{ in } \mathbb {R} \right\}$.

If $K$ is a compact subset of $\mathbb {C}$, $f \in C(K)$, and $\varepsilon > 0$, show that there is a polynomial $p(z, \bar{z})$ in $z$ and $\bar{z}$ such that $\left|f(z) - p(z,\bar{z})\right| < \varepsilon$ for all $z$ in $K$.

Let $E = \left\{ x \in l^2(\mathbb {N}) : \left\| x\right\| \leq 1 \right\}$ and for $x$ in $E$ define $f(x) = \left( (1 - \left\| x\right\|^2)^{1/2}, x(1), x(2), \dots \right)$. Show that $f(E) \subseteq E$, $f$ is continuous, and $f$ has no fixed points.

Was local convexity used in the proof of the Markov-Kakutani Fixed Point Theorem 10.1 (if $K$ is a nonempty compact convex subset of a LCS $\mathscr {X}$ and $\mathscr {F}$ is a family of continuous affine maps of $K$ into itself that is abelian, then there is an $x_0$ in $K$ such that $T(x_0) = x_0$ for all $T$ in $\mathscr {F}$)?

Show that if $X$ is locally compact and $X = \bigcup_{n=1}^{\infty } F_n$, where each $F_n$ is closed in $X$, then there is an integer $n$ such that $\interior F_n \neq \emptyset$.

Let $G$ be a group and a topological space. Show that $G$ is a topological group if and only if the map of $G \times G \to G$ defined by $(x,y) \mapsto x^{-1} y$ is continuous.

Verify the statements in Examples 11.2:

Show that Theorems 11.4 and 11.5 are equivalent, where:

• Theorem 11.4: If $G$ is a compact topological group, then there is a unique positive regular Borel measure $m$ on $G$ such that (a) $m(G) = 1$; (b) if $U$ is a nonempty open subset of $G$, then $m(U) > 0$; (c) if $\Delta$ is any Borel subset of $G$ and $x \in G$, then $m(\Delta ) = m(\Delta x) = m(x \Delta ) = m(\Delta^{-1})$, where $\Delta x \equiv \left\{ ax : a \in \Delta \right\}$, $x \Delta \equiv \left\{ xa : a \in \Delta \right\}$, and $\Delta^{-1} \equiv \left\{ a^{-1} : a \in \Delta \right\}$.

• Theorem 11.5: If $G$ is a compact topological group, then there exists a unique positive linear functional $I : C(G) \to \mathbb {F}$ such that (a) $I(1) = 1$; (b) if $f \in C(G)$, $f \geq 0$, and $f \neq 0$, then $I(f) > 0$; (c) if $f \in C(G)$ and $x \in G$, then $I(f) = I(f_x) = I({}_x f) = I(f^\# )$.

Let $G$ be a locally compact group. If $m$ is a regular Borel measure on $G$, show that any two of the following properties imply the third: (a) $m(\Delta x) = m(\Delta )$ for every Borel set $\Delta$ and every $x$ in $G$; (b) $m(x \Delta ) = m(\Delta )$ for every Borel set $\Delta$ and every $x$ in $G$; (c) $m(\Delta ) = m(\Delta^{-1})$ for every Borel set $\Delta$.

Show that the maps $S_0, L_x, R_x$ are linear isometries of $M(G)$ onto $M(G)$.

Prove (11.7): if $e =$ the identity of $G$, then

for $x, y$ in $G$.

Let $S$ be an abelian semigroup and show that there is a positive linear functional $L : l^\infty (S) \to \mathbb {F}$ such that (a) $L(1) = 1$, (b) $L(f_x) = L(f)$ for every $f$ in $l^\infty (S)$.

If $S = \mathbb {N}$, what does Exercise 7 say about Banach limits?

If $G$ is a compact group, $f : G \to \mathbb {F}$ is a continuous function, and $\varepsilon > 0$, show that there is a neighborhood $U$ of the identity in $G$ such that $\left|f(x) - f(y)\right| < \varepsilon$ whenever $xy^{-1} \in U$. (Note that this says that every continuous function on a compact group is uniformly continuous.)

If $G$ is a locally compact group and $f \in C_b(G)$, let $\mathcal{O}(f) \equiv$ the closure of $\left\{ f_x : x \in G \right\}$ in $C_b(G)$. Let $AP(G) = \left\{ f \in C_b(G) : \mathcal{O}(f) \text{ is compact} \right\}$. Functions in $AP(G)$ are called almost periodic.

Suppose $\mathscr {X}$ is a normed space and that the only hyperplanes $\mathscr {M}$ in $\mathscr {X}^*$ such that $\mathscr {M} \cap \ball \mathscr {X}^*$ is weak-star closed are those that are weak-star closed. Prove that $\mathscr {X}$ is a Banach space.

(von Neumann) Let $A$ be the subset of $l^2$ consisting of all vectors $\left\{ x_{mn} : 1 \leq m < n < \infty \right\}$ where $x_{mn}(m) = 1$, $x_{mn}(n) = m$, and $x_{mn}(k) = 0$ if $k \neq m, n$. Show that $0 \in \text{wk} - \cl A$ but no sequence in $A$ converges weakly to $0$.

Where were the hypotheses of the separability and completeness of $\mathscr {X}$ used in the proof of Theorem 12.11 (if $\mathscr {X}$ is a separable Banach space and $\mathscr {M}$ is a linear manifold in $\mathscr {X}^*$, then the following statements are equivalent: (a) $\mathscr {M}$ is weak-star sequentially dense in $\mathscr {X}^*$; (b) there is a positive constant $c$ such that for every $x$ in $\mathscr {X}$, $\left\| x\right\| \leq \sup \left\{ \left|\left\langle x, x^* \right\rangle \right| : x^* \in \mathscr {M}, \ \left\| x^*\right\| \leq c \right\}$; (c) there is a positive constant $c$ such that if $x^* \in \ball \mathscr {X}^*$, there is a sequence $\left\{ x_k^* \right\}$ in $\mathscr {M}$, $\left\| x_k^*\right\| \leq c$, such that $x_k^* \to x^*$ (wk*))?

Let $\mathscr {X}$ be a separable Banach space. If $\mathscr {M}$ is a linear manifold in $\mathscr {X}^*$, give necessary and sufficient conditions that every functional in $\text{wk}^* - \cl \mathscr {M}$ be the wk* limit of a sequence from $\mathscr {M}$.

Let $\mathscr {X}$ be a normed space and let $\mathscr {T}$ be a locally convex topology on $\mathscr {X}$ such that $\ball \mathscr {X}$ is $\mathscr {T}$-compact. Show that there is a Banach space $\mathscr {Y}$ such that $\mathscr {X}$ is isometrically isomorphic to $\mathscr {Y}^*$.

If $X$ is a compact connected topological space that is not a singleton, show that $C(X)$ is not the dual of a Banach space. (For $C_{\mathbb {R}}(X)$ this is a consequence of Exercise 7.4. For the complex case, show that if $C(X)$ is a dual space, then $C_{\mathbb {R}}(X)$ is a weak* closed real linear subspace of $C(X)$.)

Prove Corollary 13.2: if $\mathscr {X}$ is a Banach space and $A \subseteq \mathscr {X}$, then $A$ is weakly compact if and only if $A \cap \mathscr {M}$ is weakly compact for every separable subspace $\mathscr {M}$ of $\mathscr {X}$.

If $\mathscr {X}$ is a Banach space and $K$ is a compact subset of $\mathscr {X}$, prove that $\overline{\co }(K)$ is compact. (This will be proved in Theorem VI.4.8 below.)

In the proof of (13.4) (where $K$ is a weakly compact subset of a Banach space $\mathscr {X}$, $M(K) = C(K)^*$, and $T : M(K) \to \mathscr {X}$ is defined by $T(\mu ) = x_\mu$, the element of $\mathscr {X}$ such that $\left\langle x_\mu , x^* \right\rangle = \int_K \left\langle x, x^* \right\rangle \, d\mu (x)$ for all $x^*$ in $\mathscr {X}^*$), if $\mathscr {P} =$ the probability measures on $K$, show that $T(\mathscr {P}) = \overline{\co }(K)$.

Prove the Eberlein-Smulian Theorem in the setting of Hilbert space: if $\mathscr {H}$ is a Hilbert space and $A \subseteq \mathscr {H}$, then the following statements are equivalent: (a) each sequence of elements of $A$ has a subsequence that is weakly convergent; (b) each sequence of elements of $A$ has a weak cluster point; (c) the weak closure of $A$ is weakly compact.

Refer to the notation of Exercise 11.10. Prove that there is a bounded linear functional $L : AP(G) \to \mathbb {F}$ such that $L(1) = 1$, $L(f) \geq 0$ if $f \geq 0$, and $L(f_x) = L(f)$ for all $f$ in $AP(G)$ and $x$ in $G$.