P-Norm is Norm

Theorem
P-norm is a norm on the real and complex numbers.

Proof
Let $\displaystyle K \in \set {\R, \C}$.

Norm Axiom $(N1)$
By definition:


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

Suppose $\sequence {x_n}_{n \in \N_0} \in K$.

Complex modulus of $x_n$ is real and non-negative.

Sum of non-negative real numbers is non-negative.

Power of positive real number is positive.

Zero raised to a positive power is zero.

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

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

Then:

Norm Axiom $(N2)$
Suppose, $\lambda \in K$.

Norm Axiom $(N3)$
If $\mathbf x = \sequence {0}$ and $\mathbf y = \sequence {0}$, then by $\paren {N1}$ we have equality.

If $\mathbf x + \mathbf y = \sequence {0}$ and both $\bf x$ and $\bf y$ nonvanishing, then by $\paren {N1}$ we get a strict inequality.

If $\mathbf x + \mathbf y \ne \sequence {0}$, then consider p-norm raised to the power of $p$: