Generalized Hilbert Sequence Space is Metric Space

Theorem
Let $\alpha$ be an infinite cardinal number.

Let $I$ be an indexed set of cardinality $\alpha$.

Let $\struct {H^\alpha, d_2}$ be the generalized Hilbert sequence space of weight $\alpha$.

Then:
 * $\struct {H^\alpha, d_2}$ is a metric space.