Find an approximation of the unit ball as a weak-limit of a sequence in the unit sphere

Question

Let $H$ be an infinite dimensional Hilbert space.

It is well known that the weak-closure of the unit ball is the unit sphere. But I want to prove it as basicaly as possible, using the weakly-sequential deffinition of closure, by actually finding a convering sequence of members of B.

It is clear to me that $\overline S \subset B$ since $B$ is strong-closed and convex.

For the hard and interesting part, I figured out that I could pick an orthonormal basis for $H$,$\{e_a\}_{a\in A}$, and given a $x\in B$ write it as $x=\sum_{n=1}^\infty \langle x,e_n\rangle e_n$ for $(e_n)_{n=1}^{\infty} \subset \{e_a\}_{a\in A}$.

Couldn't go any further.

Please guide me a bit, and let me know if you think that what I'm trying to do is possible. A nice hint would be welcomed :)

Thanks!

Answer 1

Fix $x\in H$ with $\|x\|\leq1$. Now define $$ x_n=\sum_{k=1}^n\langle x,e_n\rangle\,e_n\ +(1-\sum_{k=1}^n|\langle x,e_n\rangle|^2)^{1/2}e_{n+1}. $$ Then $\|x_n\|=1$ for all $n$.

For any $y\in H$, $$ \langle x-x_n,y\rangle=[\langle x,e_n\rangle-(1-\sum_{k=1}^n|\langle x,e_n\rangle|^2)^{1/2}]\,\bar y_n+\sum_{k=n+1}^\infty \langle x,e_n\rangle\bar y_n\leq2\left(\sum_{k=n}^\infty|y_n|^2\right)^{1/2}\to0 $$

Answer 2

1. Let $\{e_n\}$ be an orthonormal sequence. Show that $e_n \to 0$ weakly. (Hint: Bessel's inequality says $\sum_{n=1}^\infty |\langle y, e_n \rangle|^2 \le \|y\|^2$.) This is a very useful example to keep in mind when thinking about the weak topology.

2. Show that if $\{a_n\}$ is any bounded sequence of scalars, then $a_n e_n \to 0$ weakly.

3. Fix $x \in B$. For each $n$, find a scalar $a_n$ such that $x + a_n e_n \in S$. Verify that $\{a_n\}$ is bounded.

4. Conclude.