Hilbert Sequence Space is not Locally Compact Hausdorff Space

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 $\R$.

Then $\ell^2$ is not a locally compact Hausdorff space.

Proof

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

Let $x = \sequence {x_i} \in A$ be a point of $\ell^2$.

From Point in Hilbert Sequence Space has no Compact Neighborhood, $x$ has no compact neighborhood.

Hence the result by definition of locally compact Hausdorff space.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $36$. Hilbert Space: $3$