Hilbert Sequence Space is Second-Countable
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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: $2$