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Does anyone have any recommendations where I can learn about time dependent or Bochner spaces? I mean spaces like $L^p(0,T; H^{-1}(\Omega))$. I think one needs some knowledge of distributions, so any material that covers the relevant distribution theory would be especially appreciated.

I want to get good at using these spaces and understanding the embeddings of these spaces into other Bochner spaces.

Edit: please no Evans!

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Have you left the site? You should look for Banach valued integration that can be found in several books e.g. Hille and Philips and Rudin ("Functional analysis"). The, perhaps, most common Banach valued integral is the Bochner integral. – AD. Jul 17 '13 at 22:06

Zeidler's Nonlinear Functional Analysis... book, part II/A is good for this kind of information.

I would also recommend the book about Navier Stokes by Boyer and Fabrie for a modern typesetting. Which also reminds that Temam's book (freely available on his website by Googling "Temam") also contains some basics of these kind of spaces.

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