Math Problems Vault

Problem V.5.1

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

d(\mu , \nu ) = \sum _{n=0}^{\infty } 2^{-n} \left| \int _0^1 x^n \, d\mu - \int _0^1 x^n \, d\nu \right|.

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

?

Problem V.5.2

(a)(b)(c)

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:

?

(a)

the linear span of $\left\{ f_u : u \in \mathcal{U}\right\}$ is dense in $C(X)$ ;

(b)

if $X$ is a metric space, then $C(X)$ is separable;

(c)

if $X$ is a $\sigma$ -compact metrizable locally compact space, then $C_0(X)$ is separable. ( $X$ is $\sigma$ -compact if $X$ is the union of a countable number of compact subsets.)

Problem V.5.3

(a)(b)

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

?

(a)

$\mathcal{X}$ is separable;

(b)

if $K$ is a weakly compact subset of $\mathcal{X}$ , then $K$ with the relative weak topology is metrizable.

Problem V.5.4

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

d(\phi , \psi ) = \sum _{j=1}^{\infty } 2^{-j} |\phi (j) - \psi (j)|

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

?

Problem V.5.5

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.

?

Problem V.5.6

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

?