Could any even exponent be used when calculating a norm?

Question

I know that when calculating the norm of a vector, the squared value is often used so that negative values do not affect the result. Does this mean that any even exponent may be used?

Answer 1

Yes and no.

The norm of a Euclidean vector - a coordinate vector in Euclidean ("ordinary") space - is...

$$\Vert \mathbf{x}\Vert=\sqrt{\sum_i x_i^2}$$

However, this is not the case for normed vector spaces in general. Other norms include the taxicab norm, and $L^p$-norms.

The $L^p$-norm of a vector $\mathbf{x}\in\Bbb{R}^n$ is defined for a real number $p\ge1$ as...

$$\Vert \mathbf{x}\Vert_p=\left(\sum_i|x_i|^p\right)^\frac1p$$

You can see that $\Vert\mathbf{x}\Vert_2$ is equal to the $n$-dimensional Euclidean norm. Additionally, since all norms must satisfy the triangle inequality, $\Vert\mathbf{x}\Vert_p$ is not a norm when $0<p<1$.

It is important to note that different norms on the same space are not usually equivalent. For example, $\Vert\mathbf{x}\Vert_2\ne\Vert\mathbf{x}\Vert_4$ unless $\mathbf{x}$ is a scalar multiple of a basis vector.