$( \Omega,F, P )$: a measurable space equipped with a finite measure

$(B , \Vert \cdot \Vert) $ : a Banach space with $\mathcal{B}$ as its borelian $\sigma$-algebra

$p$ : a constant bigger than $1$

Define $L^p(\Omega, B )$ the vector space that contain all $( F, \mathcal{B})$-measurable function $f$ such that :

$ \vert \Vert f \Vert \vert = \sqrt[p]{ \int_{\Omega} \Vert f \Vert ^p \cdot dP } < \infty$

I'm looking for a version of Riesz-Fischer theorem which affirms that:

Proposition: $\left( L^p(\Omega, B ) , \vert \Vert \cdot \Vert \vert \right)$ is a Banach space

With some quick calculations, I have the feeling that this proposition is quite easy to be proved. But as we all know, it's always better to have a reliable reference.

So my question is: "Is the above proposition true? And does anyone have references to this matter?"