Given any countable collection of non-zero vectors in a Hilbert space

Question

Let $\{\alpha_i\}$ be a countable collection of non-zero vectors in a Hilbert space $H$. Is there exist a vector $\beta \in H$ such that $\langle \beta , \alpha_i \rangle \neq 0$ for all $i$ ?

Answer 1

You're asking whether the union of a countable collection of orthogonal compliments of vectors can be the whole Hilbert space. The answer is "no", by the Baire category theorem, for instance. So the answer to your question is "yes", such an element always exists.