P-Sequence Space of Real Sequences is Metric Space

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Theorem

Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\ds \sum_{i \mathop \ge 0} x_i^2$ is convergent.

Let $d_p$ be the $p$-sequence metric on $\R$.


Then $\ell^p := \struct {A, d_p}$ is a metric space.


Proof

By definition of the $p$-sequence metric on $\R$:

Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\ds \sum_{i \mathop \ge 0} x_i^2$ is convergent.

Then $\ell^p := \struct {A, d_2}$ where $d_p: A \times A: \to \R$ is the real-valued function defined as:

$\ds \forall x = \sequence {x_i}, y = \sequence {y_i} \in A: \map {d_p} {x, y} := \paren {\sum_{k \mathop \ge 0} \size {x_k - y_k}^p}^{\frac 1 p}$


Proof of Metric Space Axiom $(\text M 1)$

\(\ds \map {d_p} {x, x}\) \(=\) \(\ds \paren {\sum_{k \mathop \ge 0} \size {x_k - x_k}^p}^{\frac 1 p}\) Definition of $d_p$
\(\ds \) \(=\) \(\ds \paren {\sum_{k \mathop \ge 0} 0^p}^{\frac 1 p}\)
\(\ds \) \(=\) \(\ds 0\)

So Metric Space Axiom $(\text M 1)$ holds for $d_p$.

$\Box$


Proof of Metric Space Axiom $(\text M 2)$: Triangle Inequality

Let $z = \sequence {z_i}\in A$.


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\(\ds \map {d_p} {x, y} + \map {d_p} {y, z}\) \(=\) \(\ds \paren {\sum_{k \mathop \ge 0} \size {x_k - y_k}^p}^{\frac 1 p} + \paren {\sum_{k \mathop \ge 0} \size {y_k - z_k}^p}^{\frac 1 p}\) Definition of $d_p$
\(\ds \) \(\ge\) \(\ds \paren {\sum_{k \mathop \ge 0} \size {x_k - z_k}^p}^{\frac 1 p}\) Minkowski's Inequality for Sums
\(\ds \) \(=\) \(\ds \map {d_p} {x, z}\) Definition of $d_p$

So Metric Space Axiom $(\text M 2)$: Triangle Inequality holds for $d_p$.

$\Box$


Proof of Metric Space Axiom $(\text M 3)$

\(\ds \map {d_p} {x, y}\) \(=\) \(\ds \paren {\sum_{k \mathop \ge 0} \size {x_k - y_k}^p}^{\frac 1 p}\) Definition of $d_2$
\(\ds \) \(=\) \(\ds \paren {\sum_{k \mathop \ge 0} \size {y_k - x_k}^p}^{\frac 1 p}\) Definition of Absolute Value
\(\ds \) \(=\) \(\ds \map {d_p} {y, x}\) Definition of $d_p$

So Metric Space Axiom $(\text M 3)$ holds for $d_p$.

$\Box$


Proof of Metric Space Axiom $(\text M 4)$

\(\ds x\) \(\ne\) \(\ds y\)
\(\ds \leadsto \ \ \) \(\ds \exists k \in \N: \, \) \(\ds x_k\) \(\ne\) \(\ds y_k\)
\(\ds \leadsto \ \ \) \(\ds \size {x_k - y_k}^p\) \(>\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds \paren {\sum_{k \mathop \ge 0} \size {x_k - y_k}^p}^{\frac 1 p}\) \(>\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds \map {d_p} {x, y}\) \(>\) \(\ds 0\) Definition of $d_p$

So Metric Space Axiom $(\text M 4)$ holds for $d_p$.

$\blacksquare$


Sources

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