On L^p and L^q spaces

Question

I am new in PDE's, and when I read classical things, for instance wellposedness problem, uniqueness of solutions, or somethings like that, always appears new definitions for me. In this case appear a new question: let $\Omega\subset \mathbb{R}$ and $T>0$, what is the definition of $$ L^p(0,T; L^q(\Omega)) ?$$ How we can define its norm ? what is the situation if $p=\infty$ ?

Answer 1

This is the space of almost everywhere equivalence classes of measurable maps $f:(0,T)\to L^q(\Omega) $ such that $$ \|f\|_{L^p([0,T];L^q)}:=\left(\int_0^T \|f(t)\|_q^p\textrm{d}t\right)^{\frac{1}{p}}<\infty $$ This is, of course, also the norm on this space.

In general, if $X$ is a normed space and $(E,\mu)$ is measure space, you can define $L^p(\mu;X)$ to be the set of almost everywhere equivalence classes of measurable maps $f:E\to X$ satisfying $$ \|f\|_{L^p(\mu;X)}:=\left(\int_E \|f\|^p \textrm{d}\mu\right)^{\frac{1}{p}} $$