Reference request: Simple facts about vector-valued Sobolev space

J. Wloka "Partial differential equations", § 25 (p. 390 on, in my 1992 CUP edition) has an account of the space $W(0,T)=W_2^1(0,T)$ which is essentially the space $W^{1,2}([0,T];V,V^*)$.

Herbert Ammann's book on parabolic problems contains an excellent introduction.

If you read French then this book is the place you are looking for

Brézis, H. Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. (French) North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. vi+183 pp.Inc.,

Another source

Barbu, Viorel(R-IASIM) Nonlinear differential equations of monotone types in Banach spaces. Springer Monographs in Mathematics. Springer, New York, 2010. x+272 pp. ISBN: 978-1-4419-5541-8978-1-4419-5541-8