# P-Sequence Space of Real Sequences is Metric Space

## 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 $\text M 2$

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

 $\ds \map {d_p} {x, y} + \map {d_p} {y, z}$ $=$ $\ds \paren {\sum_{k \mathop \ge 0} \size {x_k - y_k}^p}^{\frac 1 p} + \paren {\sum_{k \mathop \ge 0} \size {y_k - z_k}^p}^{\frac 1 p}$ 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 $\text M 2$ 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 $d_p$.

$\blacksquare$