What does it mean the notation $L^2(0,T;U)$?

Question

I am studying second order parabolic equations, but i can't find in my book what the notation $L^2(0,T;U)$ means. Can anybody explain this to me? Thanks

Answer 1

$U$ is some other normed vector space. In this case $L^2([0,T];U)$, sometimes lazily written as $L^2(0,T;U)$, consists of functions $f$ from $[0,T]$ to $U$ such that $\int_0^T \| f(t) \|^2 dt<\infty$, where $\| \cdot \|$ is the norm on $U$.

This notation is used in, for instance, Partial Differential Equations by Evans. Most commonly the $U$ in question is an $L^p$ space or a Sobolev space. I think this may be the usage that you are seeing.