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Let $H$ be a Hilbert space. Let $\{x_i\}$ be a sequence in $H$ with the following two properties:

  • $\|x_i\|=1$ for all $i$
  • There is a fixed number $c\geq0$ such that $(x_i,x_j)=c$ whenever $i\neq j$.

Define $$ y_n:=\frac{1}{n}\sum_{i=1}^nx_i. $$ Prove that there is a $y\in H$ such that $y_n$ converges strongly to $y$ and $x_i$ converges weakly to $y$.

I can prove that $y_n$ converges strongly to $y$ by proving it is a Cauchy sequence. But I don't know how to prove $x_i$ converges to $y$ weakly.

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Just test whether $$\langle y,v\rangle = \lim_{n\to\infty} \langle x_n,v\rangle$$ holds for every $v\in H$. This can be done by 2 steps. You test it for $v \in \text{span}\{x_j\;|\;j\geq 1\}$ first, and then for $v \in \text{span}^\perp\{x_j\;|\;j\geq 1\}$. Since $$H = \overline{\text{span}}\{x_j\;|\;j\geq 1\} \oplus \text{span}^\perp\{x_j\;|\;j\geq 1\},$$ this will prove the result.

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  • $\begingroup$ I see that. Thanks a lot! $\endgroup$ – whereamI Dec 7 '18 at 3:57

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