Compact Subset of Hilbert Sequence Space is Closed

Theorem

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

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

Let $H$ be a compact subset of $\ell^2$

Then $H$ is closed in $\ell^2$.

Proof

From Metric Space fulfils all Separation Axioms, $\ell^2$ is a Hausdorff space.

The result follows from Compact Subspace of Hausdorff Space is Closed.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $36$. Hilbert Space: $4$