# Hilbert Sequence Space is Separable

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## Theorem

Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\displaystyle \sum_{i \mathop \ge 0} x_i^2$ is convergent.

Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$.

Then $\ell^2$ is a separable space.

## Proof

Consider the set $H$ of all points of $\ell^2$ which have finitely many rational coordinates and all the rest zero.

Then $H$ forms a countable subset of $A$ which is (everywhere) dense.

The result follows by definition of separable space.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 36: \ 2$