Generalized Hilbert Sequence Space is Metric Space/Metric Space Axiom M3
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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 Metric Space Axiom $(\text M 3)$.
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 $\sequence {x_n}$ such that the series $\ds \sum_{n \mathop = 0}^\infty x_n^2$ 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:
- $\ds \forall x = \sequence {x_n}, y = \sequence {y_n} \in \ell^2: \map {d_{\ell^2}} {x, y} := \paren {\sum_{n \mathop = 0}^\infty \paren {x_n - y_n}^2}^{\frac 1 2}$
Lemma
Let $x_1, x_2, \ldots, x_m \in A$.
Then there exists $y_1, y_2, \ldots, y_m \in \ell^2$:
- $\forall a,b \in \closedint 1 m : y_a \ne y_b \iff x_a \ne x_b$
- $\forall a,b \in \closedint 1 m : \map {d_{\ell^2} } {y_a, y_b} = \map {d_2} {x_a, x_b}$
$\Box$
Let $x_1, x_2 \in A$.
From Lemma:
- $\exists y_1, y_2 \in \ell^2 : $
- $\map {d_2} {x_1, x_2} = \map {d_{\ell^2}} {y_1, y_2}$ and $\map {d_2} {x_2, x_1} = \map {d_{\ell^2}} {y_2, y_1}$
We have:
\(\ds \map {d_2} {x_1, x_2}\) | \(=\) | \(\ds \map {d_{\ell^2} } {y_1, y_2}\) | Lemma | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {d_{\ell^2} } {y_2, y_1}\) | Metric Space Axiom $(\text M 3)$ applied to $d_{\ell^2}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {d_2} {x_2, x_1}\) | Lemma |
The result follows.
$\blacksquare$