Definition of $L^p(\Omega ; H)$?

Question

I am not sure I understand the definition of the space $L^p(\Omega ; H)$. I am studying Malliavin Calculus, and the derivative operator $D$ is said to be closable from $L^p(\Omega)$ to $L^p(\Omega,H)$. $H$ being a real separable Hilbert Space.

I understand that $L^p(\Omega)$ is the set of random variables $X$ such that $$ \left(\int_\Omega| X(\omega)|^pd\omega\right)^{\frac{1}{p}} < \infty $$ but i don't understand what is $L^p(\Omega ; H)$.

Thank you for your answer!

Answer 1

If $F \in L^p(\Omega)$ and $F=f(W(h_1),\cdots, W(h_n))$, with $h_i \in H$ and good conditions on $f$, then \begin{align} DF= \sum_{i=1}^n \partial_if(W(h_1),\cdots, W(h_n))h_i \end{align}

$F$ is a real valued random variable and $DF$ takes its value in $H$. So here $L^p(\Omega, H)$ are all $H-$valued random varaible $Y$ such that $\int_{\Omega}||Y(\omega)||^p_H d\omega <+\infty$, where $||\cdot||_H$ is the norm in $H$