What mean $L^{k}(0,\infty ; H^{1}_{0}(\Omega))$

Question

Someone can explain me, what means

$L^{k}(0,\infty ; H^{1}_{0}(\Omega))$, where $k \in \mathbb{N}$,

or,

$L^{k}(0,\infty ; L^{1}(\Omega))$

Answer 1

I would read it that these are functions over $[0,\infty)$ with values in some Banach space $E$ that are $L^k$, that is, $f\in L^k([0,∞);E)$ if it is measurable in the sense that for all functionals $\alpha\in E'$ the scalar functions $α\circ f$ are measurable, and $$ \int_0^\infty\|f(t)\|_E^k\,dt<\infty. $$