Definition of $L^p$ norm of a vector-valued function

Question

If $u$ is a vector the definition of the discrete norm will be $$\|u\|_{l^p}=(\sum |u_i|^p)^{1/p},$$ If $u$ is a function, $$\|u\|_{L^p}=\left(\int|u|^p\right)^{1/p}$$ But when $u$ is a vector-valued function, what is the definition of $\|u\|_{L^p}$?

Should we use the definition of norm for each element of $u$ or should we first use the definition of discrete norm then use continuous norm definition?

Answer 1

I guess that by "$u$ is a vector", you mean "$u$ is a vector-valued function", right?

In this case, indeed the definition would be to use the definition of $l^p$ norm inside the integral term.

Answer 2

In physics vector-valued $L^{2}$-functions occur frequently. As an example let $f$ be a function from $\mathbb{R}^{n}$ into the vector space $V$ with $% |.|_{V}$ the norm on $V$. Then we denote the $L^{p}$-space as $L^{p}(\mathbb{% R}^{n},V,d^{n}x)$ or something similar. Now \begin{equation*} \parallel f\parallel _{p}^{p}=\int d^{n}x|f(x)|_{V}^{p} \end{equation*} In the common case of complex-valued $f$ we thus have $L^{p}(\mathbb{R}^{n}, \mathbb{C},d^{n}x)$ where $\mathbb{C}$is usually omitted.