Norm $L^2$ of a multivariate function

Question

I know that the $2$-norm of a function $f(x)$ of one real variable is:

$$||f(x)||_2 = \bigg (\int_x f(x)^2 dx \bigg )^{\frac{1}{2}}$$

How does this extend to a multivariate function $f : \mathbb{R}^n \rightarrow \mathbb{R}$?

Answer 1

In general, if $f : \mathbb{R}^n \rightarrow \mathbb{R}$ is a measurable map, we define

$$\|f\|_2 := \left( \int_{\Bbb R^n} |f(x_1, \dots, x_n)|^2 dx_1 \cdots dx_n \right)^{1/2}$$

This can be generalized to the $p$-norm for any $1≤p<\infty$, so that we can define the spaces $L^p(\Bbb R^n)$.