Usually, an orthonormal basis for Hilbert space means an orthonormal subset which unconditionally spans the whole space.
However, I'm curious whether there exists an orthonormal basis for every Hilbert space in algebraic sence. That is, if $H$ is a Hilbert space, does there exist a maximal linearly independent subset $\beta$ which is orthonormal? This would be false, but I cannot find a counterexample. What would it be? Thank you in advance.
In any infinite-dimensional Hilbert space $\mathcal H$, an orthonormal basis is not a basis in the algebraic sense. Suppose not: let $B$ be such an orthonormal basis. Let $b_1, b_2, \ldots$ be a sequence of distinct members of $B$ and $x = \sum_{j=1}^\infty b_j/j$. If $x = \sum_{b \in B} c_b b$ (where only finitely many $c_b \ne 0$), then $c_b = \langle b, x \rangle$. But $c_{b_j} = 1/j \ne 0$, contradiction.