Are there any books that has a nice introduction to Bochner space including its properties and proofs? Not Evans PDEs. One of my friend recommended me this: https://books.google.co.uk/books?id=peZHAAAAQBAJ&pg=PA22&lpg=PA22&dq=roubicek%20bochner&source=bl&ots=9lqwYy1AYI&sig=g1P9cx7LxqZks-5o95CoavIOzp8&hl=en&sa=X&ved=0ahUKEwjcjNWFwu3TAhUmDcAKHYPiC3wQ6AEIIzAA#v=onepage&q=roubicek%20bochner&f=false but it has statements without proofs mostly ):
2026-03-26 16:02:07.1774540927
Reference for Bochner space.
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I think (one of) the best reference is the book "Vector Measures" by Diestel and Uhl. It introduces all the concepts of weak measurability and carefully proves all the concepts, also the basic properties like the separability of $L^p(0,T;X)$ if $X$ is separable, a dominated convergence theorem in Bochner spaces and so on.
PDEs books which treat evolution equations also introduce Bochner spaces, but in a different way. Just think of the books by Renardy&Rogers, Evans, Salsa, Roubicek. They all shortly state on ~5 pages the results they need but they do not prove (most of) them. So I think a good reference is the book above by Diestel&Uhl, it is also a nice read in my opinion.