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 $H^\alpha$ be the generalized Hilbert sequence space of weight $\alpha$.

Then:
 * $H^\alpha$ is a metric space.

Proof
Recall $H^\alpha$ is the structure $\struct{A, d_2}$ where:
 * $A$ is the set of all real-valued functions $x : I \to \R$ such that:
 * $(1)\quad \set{i \in I: x_i \ne 0}$ is countable
 * $(2)\quad$ the generalized sum $\ds \sum_{i \mathop \in I} x_i^2$ is a convergent net.
 * $d_2: A \times A: \to \R$ is the real-valued function defined as:
 * $\ds \forall x = \family {x_i}, y = \family {y_i} \in A: \map {d_2} {x, y} := \paren {\sum_{i \mathop \in I} \paren {x_i- y_i}^2}^{\frac 1 2}$

Let $H = \struct{\ell^2, d_{\ell^2}}$ denote the Hilbert sequence space, where:
 * $\ell^2$ denotes the real $2$-sequence space, that is, the set of all real sequences $\sequence {x_i}$ such that the series $\ds \sum_{i \mathop = 0}^\infty x_i^2$ is convergent
 * $d_{\ell^2}$ denotes the real-valued function $d_{\ell^2}: \ell^2 \times \ell^2: \to \R$ defined as:
 * $\ds \forall x = \sequence {x_i}, y = \sequence {y_i} \in \ell^2: \map {d_{\ell^2}} {x, y} := \paren {\sum_{k \mathop \ge 0} \paren {x_k - y_k}^2}^{\frac 1 2}$