Generalized Hilbert Sequence Space is Metric Space/Metric Space Axiom M3

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

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

Let $A$ be 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.

Let $d_2: A \times A: \to \R$ be the real-valued function defined as:
 * $\ds \forall x = \family {x_i}, y = \family {y_i} \in A: \map {d_2} {x, y} := \paren {\sum_{i \mathop \in I} \paren {x_i- y_i}^2}^{\frac 1 2}$

Then $d_2$ satisfies.

Proof
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 $2$-sequence metric, that is, the real-valued function $d_{\ell^2}: \ell^2 \times \ell^2: \to \R$ defined as:
 * $\ds \forall x = \sequence {x_i}, y = \sequence {y_i} \in \ell^2: \map {d_{\ell^2}} {x, y} := \paren {\sum_{k \mathop \ge 0} \paren {x_k - y_k}^2}^{\frac 1 2}$

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

Let $\ds \sum_{i \mathop \in I} x_i^2, \sum_{i \mathop \in I} y_i^2$ and $\ds \sum_{i \mathop \in I} z_i^2$ converge to $a, b, c \in \R$ respectively.

From lemma 2, there exists enumeration $J = \set{j_0, j_1, j_2, \ldots}$ of a countable set of $I$:


 * $\set{i \in I : x_i \ne 0}, \set{i \in I : y_i \ne 0}, \set{i \in I : y_i \ne 0} \subseteq J$


 * $\sequence{x_{j_k}}, \sequence{y_{j_k}}, \sequence{z_{j_k}} \in \ell^2$


 * $\ds \sum_{k \mathop = 0}^\infty x_{j_k}^2 = a, \sum_{k \mathop = 0}^\infty y_{j_k}^2 = b, \sum_{k \mathop = 0}^\infty z_{j_k}^2 = c$

From P-Sequence Space with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space:
 * $\sequence{x_{j_k} - y_{j_k}} \in {\ell^2}$


 * $\sequence{x_{j_k} - z_{j_k}} \in {\ell^2}$


 * $\sequence{z_{j_k} - y_{j_k}} \in {\ell^2}$

We have:
 * $\forall i \in I \setminus J : x_i = y_i = 0$

Hence:
 * $\forall i \in I \setminus J : x_i - y_i = 0$

It follows:
 * $\set{i : x_i - y_i \ne 0} \subseteq J$

Similarly:
 * $\set{i : x_i - z_i \ne 0} \subseteq J$
 * $\set{i : z_i - y_i \ne 0} \subseteq J$

From lemma 1:
 * $\ds \sum_{i \mathop \in I} \paren{x_i - y_i}^2 = \sum_{k \mathop = 0}^\infty \paren{x_{j_k} - y_{j_k}}^2 < \infty$
 * $\ds \sum_{i \mathop \in I} \paren{x_i - z_i}^2 = \sum_{k \mathop = 0}^\infty \paren{x_{j_k} - z_{j_k}}^2 < \infty$
 * $\ds \sum_{i \mathop \in I} \paren{z_i - y_i}^2 = \sum_{k \mathop = 0}^\infty \paren{z_{j_k} - y_{j_k}}^2 < \infty$

We have: