Hilbert Sequence Space is Lindelöf
Jump to navigation
Jump to search
Theorem
Let $\ell^2$ be the Hilbert sequence space on $\R$.
Then $\ell^2$ is a Lindelöf space.
Proof
From Hilbert Sequence Space is Second-Countable, $\ell^2$ is a second-countable space.
The result follows from Second-Countable Space is Lindelöf.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $36$. Hilbert Space: $2$