Solution to Problem of the Week 4

So, this is really late, the semester took its toll on me, and again nothing works.

The first problem is an application of the Eberlain-Smulyan theorem; let ${x_k}_{k\in\mathbb{N}}$ be a countable dense subset of $X$ and note that for each $k\in \mathbb{N}$ there exists a subsequence $\Lambda_k\subset \mathbb{N}$ such that $(F_n(x_k))_{n\in\Lambda_k}$ is weakly convergent. We choose the subsequences so that $\Lambda_1\subset\Lambda_2\subset\dots$ and form a diagonal sequence from these and call it $\Lambda$ .Let $l\in Y^*$ and $x\in X$ and let $\eps>0$ . Choose $k\in \mathbb{N}$ large enough so that

$||x-x_k||_X<\eps$

and $n,m\in\Lambda$ large enough so that

$|l(F_n(x_k))-l(F_m(x_k))|< \eps.$

Then it follows that

$\abs{l(F_n(x))-l(F_m(x))}\leq \eps.$

The trick here was to use the $\eps$ definition to write it out rigorously, hence bypassing the need for dealing with limits.

For the second problem:
We first take a subsequence which we do not re-label so that $F(x_k)\to\alpha_0$ where $\alpha_0:=\liminf_{k\to\infty}F(x_k)\leq - \infty$ . We then use Mazur’s lemma to write down a convex combination of the $(x_k)_{k\geq k_0}$ , call them $(y_k)_{k\geq k_0}$ that converge strongly to $x$ for every $k_0\in\mathbb{N}$ . This is important that we create these sequences truncating off the first $k_0$ elements, since then this will allow us to pass the limit in the inequalities we will later obtain. More precisely, there exists $0 \leq \alpha_{k,l} \leq 1$ with $\sum_{l=k_0} \alpha_{k,l} = 1$ such that $y_k :=\sum_{l\in\mathbb{N}}\alpha_{k,l}x_l$ converges strongly to $x$ and so for every $k\geq k_0$ ,