Characterization of Generalized Hilbert Sequence Space

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

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

Let $H^\alpha = \struct{A, d_2}$ be the generalized Hilbert sequence space of weight $\alpha$ where:
 * $A$ denotes 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 $H = \struct{\ell^2, d_{\ell^2}}$ 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 $\ds \sum_{n \mathop = 0}^\infty x_n^2$ is convergent

Let $x_1, x_2, \ldots, x_m : I \to \R$ be real-valued functions.

Then:
 * $x_1, x_2, \ldots, x_m \in A$


 * there exists an enumeration $\set{j_0, j_1, j_2, \ldots}$ of a countably infinite subset of $I$ such that $\forall k \in \closedint 1 m$:
 * $(1)\quad\set{i \in I : \paren{x_k}_i \ne 0} \subseteq \set{j_0, j_1, j_2, \ldots}$


 * $(2)\quad\sequence{\paren{x_k}_{j_n}} \in \ell^2$

In which case:
 * $k \in \closedint 1 m : \ds \sum_{i \mathop \in I} \paren{\paren{x_k}_i}^2 = \sum_{n \mathop = 0}^\infty \paren{\paren{x_k}_{j_n}}^2$

Necessary Condition
Let $x_1, x_2, \ldots, x_m \in A$.

By definition of $A$:
 * $\forall k \in \closedint 1 m : \set{i \in I : \paren{x_k}_i \ne 0}$ is countable

From Infinite Set has Countably Infinite Subset, let:
 * $I' \subseteq I$ be countably infinite

Let:
 * $J = I' \cup \ds \bigcup_{k \mathop = 1}^m \set{i \in I : \paren{x_k}_i \ne 0}$

From Countable Union of Countable Sets is Countable:
 * $J$ is countable

From the contrapositive statement of Subset of Finite Set is Finite:
 * $J$ is countably infinite

From User:Leigh.Samphier/Topology/Countably Infinite Set has Enumeration, let:
 * $\set{j_0, j_1, j_2, \ldots}$ be an enumeration of $J$

From Set is Subset of Union:
 * $\forall k \in \closedint 1 m : \set{i \in I : \paren{x_k}_i \ne 0} \subseteq \set{j_0, j_1, j_2, \ldots}$

By definition of $A$:
 * $\forall k \in \closedint 1 m : \ds \sum_{i \mathop \in I} \paren{x_k}_i^2$ converges

From Square of Real Number is Non-Negative and : $\forall k \in \closedint 1 m : \forall i \in I : \paren{x_k}_i^2 = \size{\paren{x_k}_i^2}$

Hence:
 * $\forall k \in \closedint 1 m : \ds \sum_{i \mathop \in I} \size{\paren{x_k}_i^2}$ is convergent

From User:Leigh.Samphier/Topology/Generalized Sum with Countable Non-zero Summands:
 * $\forall k \in \closedint 1 m : \ds \sum_{n \mathop = 0}^\infty \size{\paren{\paren{x_k}_{j_n}}^2}$ is convergent

and
 * $\forall k \in \closedint 1 m : \ds \sum_{i \mathop \in I} \size{\paren{x_k}_i^2} = \sum_{n \mathop = 0}^\infty \size{\paren{\paren{x_k}_{j_n}}^2}$

Since for all $k \in \closedint 1 m, i \in I : {\paren{x_k}_i}^2 = \size{\paren{x_k}_i^2}$, then:
 * $\forall k \in \closedint 1 m$:
 * $\ds \sum_{n \mathop = 0}^\infty \paren{\paren{x_k}_{j_n}}^2$ is convergent, that is, $\sequence{\paren{x_k}_{j_n}} \in \ell^2$
 * and
 * $\ds \sum_{i \mathop \in I} \paren{x_k}_i^2 = \sum_{n \mathop = 0}^\infty \paren{\paren{x_k}_{j_n}}^2$

Sufficient Condition
Let $\set{j_0, j_1, j_2, \ldots}$ be a countably infinite subset of $I$:


 * $(1)\quad \set{i \in I : x_i \ne 0} \subseteq \set{j_0, j_1, j_2, \ldots}$


 * $(2)\quad \sequence{x_{j_n}} \in \ell^2$

By definition of $\ell^2$:
 * $\ds \sum_{n \mathop = 0}^\infty \paren{x_{j_n}}^2 < \infty$

From Square of Real Number is Non-Negative and :
 * $\forall i \in I : {x_i}^2 = \size{ x_i^2}$

Hence:
 * $\ds \sum_{n \mathop = 0}^\infty \size{\paren{x_{j_n}}^2}$ is convergent

From User:Leigh.Samphier/Topology/Generalized Sum with Countable Non-zero Summands:
 * $\ds \sum_{i \mathop \in I} \size{x_i^2}$ is a convergent net

and
 * $\ds \sum_{i \mathop \in I} \size{x_i^2} = \sum_{n \mathop = 0}^\infty \size{\paren{x_{j_n}}^2}$

Since for all $i \in I : {x_i}^2 = \size{ x_i^2}$, then:
 * $\ds \sum_{i \mathop \in I} x_i^2$ is a convergent net

and
 * $\ds \sum_{i \mathop \in I} x_i^2 = \sum_{n \mathop = 0}^\infty \paren{x_{j_n}}^2$