# Proving a weak convergence theorem using ascoli's theorem proof

Theorem: Let $$(H, \langle\cdot,\cdot\rangle)$$ be a separable Hilbert space. Let $$(x_n)_{n\in \mathbb{N}}$$ be a sequence such that $$x_n\in H$$ and $$\|x_n\|\leqslant 1$$ for all $$n\in \mathbb{N}$$. Then, there exist a subsequence $$(x_{j(n)})$$ and $$x\in H$$ such that $$x_{j(n)} \rightarrow x \mbox{ weakly in }\ H.$$ To prove this theorem, I need to do the following steps:

1. For $$n\in \mathbb{N}$$, we suppose $$f_n: H \rightarrow \mathbb{R}$$ defined by $$f_n(z)=\langle x_n, z\rangle$$ for all $$z\in H$$. Show that the weak convergence will return to show the simple convergence of f with close extraction.

2. Taking the inspiration from the proof of Ascoli's theorem, we will show that there exists $$f ∈ H'$$ and a subsequence $$(f_{j(n)})$$ such that $$(f_{j(n)})$$ converges simply to $$f$$. There will be a small diagional process of Cantor. It will be simpler than Ascoli actually.

3. We will go back to H with a well-chosen theorem.

What we need here is to prove this theorem, but for that we will prove the fthree steps mentioned above. I face a problem in find a solution for this theorem. Does anyone can help me?

• Are you looking for something like this? – Thomas Shelby Nov 30 '19 at 15:43
• @ThomasShelby thank you for the comment. It can help to prove the weak convergence of the subsequence $x_{j(n)}$. But what am searching on is to prove it using the above three steps. especially step 2. – user478765 Nov 30 '19 at 15:55