Reference for Bochner space. 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 ): 
 A: Besides L. Evans book you could find Bochner spaces in a more then a few books. It all depends what kind of proofs do you need, what kind of problems you are studying, etc. Bellow I will write two books that I like.
A very short introduction of Bochner spaces you could find in the book: 
Málek, Nečas, Rokyta, Růžička - Weak and Measure-valued Solutions to Evolutionary PDEs, 1996 - subsection 1.2.6
More you can find in the book:
Pascal Cherrier, Albert Milani - Linear and Quasi Linear Evolution Equations in Hilbert Spaces, 2010 - introduction in section 1.7, other different theorems  in other chapters.
Hope this helps.
A: 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.
