$u \in L^{\infty}(0,T;H^{1})$, What does this notation mean?

Question

I am trying to understand the error analysis for a mixed finite element method. I am told $u$ must be in the above space, but this notation is unclear to me.

Answer 1

This mean that $u(t), 0\leq t \leq T$ is a function, and for each fixed $t$, $u(t)\in H^1$. There may be a defined norm in $H^1$, say $||\cdot||$, then $L^\infty$ means there exists a finite $M>0$ such that $||u(t)|| < M$ for almost every $t$