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:
 * $\ds \forall x = \family {x_i}, y = \family {y_i} \in A: \map {d_2} {x, y} := \paren {\sum_{i \mathop \in I} \paren {x_i- y_i}^2}^{\frac 1 2}$

Lemma
Let $x^{\paren 1}, x^{\paren 2}, \ldots, x^{\paren k} \in A$.

Then there exists an enumeration $\set{j_0, j_1, j_2, \ldots}$ of a subset of $I$:


 * $\forall n \in \closedint 1 k$:


 * $(1)\quad \set{i \in I : x^{\paren n}_i \ne 0} \subseteq \set{j_0, j_1, j_2, \ldots}$


 * $(2)\quad \sequence{x^{\paren n}_{j_l}} \in \ell^2_\R$


 * $(3)\quad \ds \sum_{i \mathop \in I} \paren{x^{\paren n}_i}^2 = \sum_{l \mathop = 0}^\infty \paren{x^{\paren n}_{j_l}}^2$

where $\ell^2_\R$ denotes the real $2$-sequence space.

$d_2$ is Well-defined
From Characterization of Hausdorff Property in terms of Nets:
 * a convergent net in $\R$ has a unique limit.

To show that $d_2$ is well-defined, it is sufficient to show:
 * $\ds \forall x = \family {x_i}, y = \family {y_i} \in A:$ the generalized sum $\ds \sum_{i \mathop \in I} \paren {x_i- y_i}^2$ converges

Let $x = \family {x_i}, y = \family {y_i} \in A$.

Let $x, y$ converge to $a, b \in \R$ respectively.

Let $J_x = \set{i \in I : x_i \ne 0}$.

Let $J_y = \set{i \in I : y_i \ne 0}$.

Let $J_{x \mathop - y} = \set{i \in I : x_i - y_i \ne 0}$.

Let $J = J_x \cup J_y$.

From :
 * $J$ is countable

Lemma

 * $J_{x \mathop - y} \subseteq J$

From :
 * $J_{x \mathop - y}$ is countable

Let $\set{j_0, j_1, j_2, \ldots}$ be an enumeration of $J$.

Lemma

 * the series $\ds \sum_{k \mathop = 0}^\infty x_{j_k}^2$ converges to $a$


 * the series $\ds \sum_{k \mathop = 0}^\infty y_{j_k}^2$ converges to $b$

By definition of real $2$-sequence space ${\ell^2}_\R$:
 * the sequences $\sequence{x_{j_k}}, \sequence{y_{j_k}} \in {\ell^2}_\R$