Let $H$ be an infinite dimensional separable Hilbert space with orthonormal basis $(e_n)_n≥1$. Let $f_N = \frac{1}{N} \sum^{N^2}_{n=1} e_n$, $\forall N ≥ 1$.
I am trying to show that $e_n → 0$ weakly, as $n →∞$ and $f_N → 0$ weakly, as $N →∞$, while $ \|f_N\| = 1$, $\forall N ≥ 1$.
Let $x=\sum a_n e_n$ be any element of H.Then $<x,f_N>={\frac 1 N} \sum_1 ^{N^{2}} a_n$. Given that $\{a_n\} \in l^{2}$ we have to show that ${\frac 1 N} \sum_1 ^{N^{2}} a_n \to 0$. Let $\epsilon >0$. Choose m such that $\sum _m ^{\infty} {|a_n|^{2}} <\epsilon$. Since ${\frac 1 N} \sum_1 ^{m-1} a_n \to 0$ it suffices to show that ${\frac 1 N} \sum_m ^{N^{2}} a_n \to 0$. Apply Cauchy -Schwartz inequlaity: $(|\sum_m ^{N^{2}} a_n|)^{2} \leq \sum _m ^{N^{2}} |a_n|^{2} (N^{2}-m+1)< \epsilon (N^{2}-m+1))$. Divide by $n^{2}$ and take the limit.