If $A(t)$ denotes for each fixed $t$ a (smooth) surface in $\mathbb{R}^n$, what is the norm on the space $$L^2\left(\cup_{t \in [0,T]} A(t)\times \{t\}\right)?$$
Is it $$\lVert f \rVert^2 = \int_0^T{\lVert f \rVert^2_{L^2(A(t))}}$$ (the usual Bochner space thing) or is it $$\lVert f \rVert^2 = \int_0^T\int_{A(t)}{f^2(x,t)}\;\mathrm{d}x\;\mathrm{d}t$$
If it's the latter can I change the order of integration?