Closed linear subset of a Hilbert space

Question

If $H$ is a Hilbert space, and if $$(a,b)_H=0$$ for every $b \in B \subset H$, where $B$ is a closed linear subset of $H$, does it follow that $a=0$, the zero element of $H$?

Answer 1

No, because if $B\neq H$, $B$ will have a non-zero orthogonal complement. That means precisely that there exist non-zero $a\in H$ such that $(a,B)=0$.

For an extreme example, just take $B=0$ in a non-zero Hilbert space.

Answer 2

No. For example, take $H=\Bbb R^2$, $B=\{(0,a)\mid a\in \Bbb R\}$, and $a=(1,0)$.