Hilbert Sequence Space is Second-Countable

Theorem

Let $\ell^2$ be the Hilbert sequence space on $\R$.

Then $\ell^2$ is a second-countable space.

Proof

From Hilbert Sequence Space is Separable, $\ell^2$ is a separable space.

We also have that Hilbert Sequence Space is Metric Space.

The result follows from Separable Metric Space is Second-Countable.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $36$. Hilbert Space: $2$