# Prove that $x_i$ converges weakly to $y$.

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.

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.