I ran into the following exercise:
Let H be a Hilbert space. Define $U(x,\epsilon),\ S$ as follows: $$ \begin{aligned} U(x,\epsilon) &= \{y\in H: \lvert<x,y>\rvert<\epsilon \} \\ S &= \{U(x,\epsilon ):x\in H,\epsilon>0\} \end{aligned} $$ Prove that S is a subasis for the weak topology.
It's pretty easy to show that for every non-zero functional $\varphi\in H^*$, and every open ball $B(z,\epsilon)\subset \mathbb C $, there exists $x,y\in H$ such that: $$ \varphi^{-1} (B)=x+U(y,\epsilon )$$
This reduces the problem to proving that the topology generated by S is translation invariant.
I've found that difficult. Can anyone provide me with a hint of some sort?