nLab separable Hilbert space

Definition

A Hilbert space $H$ over a field $F$ of real or complex numbers and with inner product $(|)$ is separable if it has a countable topological base, i. e. a family of vectors $e_i$, $i\in I$ where $I$ is at most countable, and such that every vector $v\in H$ can be uniquely represented as a series $v = \sum_{i\in I} a_i e_i$ where $a_i\in F$ and the sum converges in the norm $\|x\| = \sqrt{(x|x)}$.

Related concepts

• separable space

• $U(\mathcal{H})$