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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?

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Surely the two norms, as you've written them, are the same! In the 2nd case, the inner integral is just $\|f\|^2_{L^2}$. My other concern is: how are the $A(t)$ varying in $t$? As written, there is no relation between them, so it's not at all clear what measure the union can be given. –  Matthew Daws Jun 14 '12 at 9:34
    
@MatthewDaws yeah you're right! The $A(t)$ are $C^{2,1}$ hypersurfaces, so locally $A(t)$ is the zero level set of a $C^{2,1}$ function $u(x,t)$. –  blahb Jun 14 '12 at 9:59

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