Characterization of Generalized Hilbert Sequence Space/Sufficient 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 $x_1, x_2, \ldots, x_m : I \to \R$ be real-valued functions.
Let there exist an enumeration $\set{j_0, j_1, j_2, \ldots}$ of a countably infinite subset of $I$ such that $\forall k \in \closedint 1 m$:
- $(1)\quad\set{i \in I : \paren{x_k}_i \ne 0} \subseteq \set{j_0, j_1, j_2, \ldots}$
- $(2)\quad\sequence{\paren{x_k}_{j_n}} \in \ell^2$
Then:
- $x_1, x_2, \ldots, x_m \in A$
In which case:
- $\forall k \in \closedint 1 m : \ds \sum_{i \mathop \in I} \paren{\paren{x_k}_i}^2 = \sum_{n \mathop = 0}^\infty \paren{\paren{x_k}_{j_n}}^2$
Proof
By definition of $\ell^2$:
- $\forall k \in \closedint 1 m : \ds \sum_{n \mathop = 0}^\infty \paren{\paren{x_k}_{j_n}}^2 < \infty$
We have:
\(\ds \forall k \in \closedint 1 m: \, \) | \(\ds \sum_{n \mathop = 0}^\infty \paren{\paren{x_k}_{j_n} }^2\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \size{\paren{\paren{x_k}_{j_n} }^2}\) | Square of Real Number is Non-Negative and Definition of Absolute Value | ||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop \in I} \size{\paren{x_k}_i^2}\) | Generalized Sum with Countable Non-zero Summands | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop \in I} \paren{x_k}_i^2\) | Generalized Sum with Countable Non-zero Summands |
$\blacksquare$