# P-Norm is Norm/P-Sequence Space

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## Theorem

The $p$-norm on the $p$-sequence space is a vector space norm.

## Proof

### Norm Axiom $\text N 1$: Positive Definiteness

Let $x \in \ell^p$ with $1 \le p < \infty$.

By definition of $p$-norm:

$\ds \norm {\mathbf x}_p = \paren {\sum_{n \mathop = 0}^\infty \size {x_n}^p}^{1/p}$

The complex modulus of $x_n$ is real and non-negative.

We have the results:

Sum of Non-Negative Reals is Non-Negative
Power of Positive Real Number is Positive
Zero Raised to Positive Power is Zero

Hence, $\norm {\mathbf x}_p \ge 0$.

Suppose $\norm {\mathbf x}_p = 0$.

Then:

 $\ds \norm {\mathbf x}_p$ $=$ $\ds 0$ $\ds \leadsto \ \$ $\ds \sum_{n \mathop = 0}^\infty \size {x_n}^p$ $=$ $\ds 0$ raising to power $p > 0$ $\ds \leadsto \ \$ $\ds \size {x_n}$ $=$ $\ds 0$ Sum of Non-Negatives vanishes iff Summands vanish $\ds \leadsto \ \$ $\ds x_n$ $=$ $\ds 0$ Complex Modulus equals Zero iff Zero $\ds \leadsto \ \$ $\ds \bf x$ $=$ $\ds \sequence 0_{n \mathop \in \N}$

Thus Norm Axiom $\text N 1$: Positive Definiteness is satisfied.

$\Box$

### Norm Axiom $\text N 2$: Positive Homogeneity

Suppose $\alpha \in \set {\R, \C}$.

 $\ds \norm {\alpha \cdot \mathbf x}_p$ $=$ $\ds \paren {\sum_{n \mathop = 0}^\infty \size {\alpha x_n}^p }^{\frac 1 p}$ $\ds$ $=$ $\ds \paren {\sum_{n \mathop = 0}^\infty \size \alpha^p \size {x_n}^p }^{\frac 1 p}$ $\ds$ $=$ $\ds \size {\alpha} \paren {\sum_{n \mathop = 0}^\infty \size {x_n}^p }^{\frac 1 p}$ $\ds$ $=$ $\ds \size {\alpha} \norm {\mathbf x}_p$

Thus Norm Axiom $\text N 2$: Positive Homogeneity is satisfied.

$\Box$

### Norm Axiom $\text N 3$: Triangle Inequality

From P-Norm is Norm we have that:

$\ds \paren {\sum_{n \mathop = 0}^d \size {x_n + y_n}^p}^{\frac 1 p} \le \paren {\sum_{n \mathop = 0}^d \size {x_n}^p }^{\frac 1 p} + \paren {\sum_{n \mathop = 0}^d \size {y_n}^p }^{\frac 1 p}$

$\map f z = z^{\frac 1 p}$ is a continuous function for $z \ge 0$ and $p > 0$.

For $\mathbf x \in \ell^p$, changing $d$ is equivalent to changing $z$ in the interval $0 \le z < \infty$.

Take the composite limit $d \to \infty$.

Then:

$\norm {\mathbf x + \mathbf y}_p \le \norm {\mathbf x}_p + \norm {\mathbf y}_p$

Thus Norm Axiom $\text N 3$: Triangle Inequality is satisfied.

$\blacksquare$