Hilbert Sequence Space is not Sigma-Compact

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 not $\sigma$-compact.

Proof
By Compact Subset of Hilbert Sequence Space is Nowhere Dense, a compact subset of $\ell^2$ is nowhere dense in $\ell^2$.

We have that Hilbert Sequence Space is Complete Metric Space.

Hence $\ell^2$ is non-meager.

It follows that $\ell^2$ is not $\sigma$-compact.