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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?

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  • $\begingroup$ Are you looking for something like this? $\endgroup$ – Thomas Shelby Nov 30 '19 at 15:43
  • $\begingroup$ @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. $\endgroup$ – user478765 Nov 30 '19 at 15:55

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