Generalized Hilbert Sequence Space is Metric Space
 | This article needs proofreading. Please check it for mathematical errors. If you believe there are none, please remove {{Proofread}} from the code.To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code. |
Theorem
Let $\alpha$ be an infinite cardinal number.
Let $I$ be an indexed set of cardinality $\alpha$.
Let $H^\alpha$ be the generalized Hilbert sequence space of weight $\alpha$.
Then:
- $H^\alpha$ is a metric space.
Proof
Recall $H^\alpha$ is the structure $\struct{A, d_2}$ where:
- $A$ is 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.
- $d_2: A \times A \to \R$ is the real-valued function defined as:
- $\ds \forall x = \family {x_i}, y = \family {y_i} \in A: \map {d_2} {x, y} := \paren {\sum_{i \mathop \in I} \paren {x_i- y_i}^2}^{\frac 1 2}$
$d_2$ is Well-defined
From Characterization of Hausdorff Property in terms of Nets:
- a convergent net in $\R$ has a unique limit.
To show that $d_2$ is well-defined, it is sufficient to show:
- $\ds \forall x = \family {x_i}, y = \family {y_i} \in A:$ the generalized sum $\ds \sum_{i \mathop \in I} \paren {x_i - y_i}^2$ converges
Let $x = \family {x_i}, y = \family {y_i} \in A$.
Let $\ds \sum_{i \mathop \in I} x_i^2$ and $\ds \sum_{i \mathop \in I} y_i^2$ converge to $a, b \in \R$ respectively.
From Characterization of Generalized Hilbert Sequence Space:
- there exists enumeration $J = \set{j_0, j_1, j_2, \ldots}$ of a countable set of $I$:
- $\set{i \in I : x_i \ne 0}, \set{i \in I : y_i \ne 0} \subseteq J$
- $\sequence{x_{j_n}}, \sequence{y_{j_n}} \in \ell^2$
- $\ds \sum_{n \mathop = 0}^\infty x_{j_n}^2 = a, \sum_{n \mathop = 0}^\infty y_{j_n}^2 = b$
- $\sequence{x_{j_n} - y_{j_n}} \in {\ell^2}$
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:
- $\set{i : x_i - y_i \ne 0} \subseteq J$
From Characterization of Generalized Hilbert Sequence Space:
- $\family{x_i - y_i}_{i \mathop \in I} \in A$
By definition of $A$:
- the generalized sum $\ds \sum_{i \mathop \in I} \paren {x_i - y_i}^2$ converges
The result follows.
$\Box$
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
- $d_{\ell^2}$ denotes 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 $y_1, y_2, \ldots, y_m \in \ell^2$:
- $\forall a,b \in \closedint 1 m : y_a \ne y_b \iff x_a \ne x_b$
- $\forall a,b \in \closedint 1 m : \map {d_{\ell^2} } {y_a, y_b} = \map {d_2} {x_a, x_b}$
$\Box$
$d_2$ satisfies Metric Space Axiom $(M1)$
Let $x \in A$.
From Lemma:
- $\exists y \in \ell^2 :$
- $\map {d_2} {x, x} = \map {d_{\ell^2}} {y, y}$
We have:
| \(\ds \map {d_2} {x, x}\) | \(=\) | \(\ds \map {d_{\ell^2} } {y, y}\) | Lemma | |||||||||||
| \(\ds \) | \(=\) | \(\ds 0\) | Metric Space Axiom $(\text M 4)$ applied to $d_{\ell^2}$ |
The result follows.
$\Box$
$d_2$ satisfies Metric Space Axiom $(M2)$
Let $x_1, x_2, x_3 \in A$.
From Lemma:
- $\exists y_1, y_2, y_3 \in \ell^2 : $
- $\forall i, j \in \set{1, 2, 3} : \map {d_2} {x_i, x_j} = \map {d_{\ell^2}} {y_i, y_j}$
We have:
| \(\ds \map {d_2} {x_1, x_3}\) | \(=\) | \(\ds \map {d_{\ell^2} } {y_1, y_3}\) | Lemma | |||||||||||
| \(\ds \) | \(\le\) | \(\ds \map {d_{\ell^2} } {y_1, y_2} + \map {d_{\ell^2} } {y_2, y_3}\) | Metric Space Axiom $(\text M 2)$: Triangle Inequality applied to $d_{\ell^2}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map {d_2} {x_1, x_2} + \map {d_2} {x_2, x_3}\) | Lemma |
The result follows.
$\Box$
$d_2$ satisfies Metric Space Axiom $(M3)$
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.
$\Box$
$d_2$ satisfies Metric Space Axiom $(M4)$
Let $x_1, x_2 \in A : x_1 \ne x_2$.
From Lemma:
- $\exists y_1, y_2 \in \ell^2 :$
- $y_1 \ne y_2$
- $\map {d_2} {x_1, x_2} = \map {d_{\ell^2}} {y_1, y_2}$
We have:
| \(\ds \map {d_2} {x_1, x_2}\) | \(=\) | \(\ds \map {d_{\ell^2} } {y_1, y_2}\) | Lemma | |||||||||||
| \(\ds \) | \(>\) | \(\ds 0\) | Metric Space Axiom $(\text M 4)$ applied to $d_{\ell^2}$ |
The result follows.
$\blacksquare$