Neighborhood of the Origin in the Weak Topology I am in difficulty with the following question. Let $H$ be a real infinite-dimensional Hilbert space and $u\in H\setminus \{0\}$. Let $V$ be an any neighborhood of $0$ in the weak topology on $H$. Is there a vector $v\ne 0$ such that $tv\in V$ for all $t\in \mathbb{R}$ and $\langle u, v\rangle\ne 0$?
I would like to thank all help and comments.

Base on the solution of Daniel Fischer, choosing $V=\{x\in H: |\langle u, x\rangle|<1\}$. 
Then $V$ is a neighborhood of $0$ in the weak topology. Let $v\ne 0$ such that $tv\in V$ for all $t\in \mathbb{R}$. Then $\langle u, tv\rangle=t\langle u, v\rangle<1$ for all $t\in \mathbb{R}$. Hence $\langle u, v\rangle=0$.
