Characterization of Generalized Hilbert Sequence Space/Necessary Condition

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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 $x_1, x_2, \ldots, x_m \in A$.

Then: 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:

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