How to define the Hilbert space

Question

Let $I:=[a,b]$ and $L^2_k(X)$ be a Hilbert space over some compact space $X$.

Q: How to define the metric/norm of $L^2(I,L^2_k(X))$?

Answer 1

$$||r|| =\sqrt{\int_a^b ||r(t)||_2^2 dt }$$