Hilbert Cube is Metric Space

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Theorem

Let $M = \left({I^\omega, d_2}\right)$ be the Hilbert cube.


Then $M$ is a metric space.


Proof

As defined, $M$ is a subspace of the Hilbert sequence space $\ell^2$.

We have that Hilbert Sequence Space is Metric Space.

The result follows from Subspace of Metric Space is Metric Space.

$\blacksquare$


Sources