I learned **basic** results (regarding weak convergence) about Banach-space valued functions of a single real variable when learning PDE. (See e.g. Appendix E in Evans's *Partial Differential Equations*) I can only find a treatment about this topic in Yosida's *Functional Analysis*, which is quite a classical old book. 


Could anyone come up with some more recent references (for an **elementary** introduction to vector-valued $L^p$ spaces) than Yosida's one?




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reference request: simple facts about vector-valued $L^p$ spaces
Too long for a comment...


If I am not mistaken, the last (=6th) edition of Yosida's Functional Analysis is from 1995, so it seems you want something not older than that. Here is a list:


1. Marsden, Ratiu, Abraham - Manifolds, Tensor Analysis, and Applications: lecture notes from 2007 available online, it has a full chapter on differential calculus in Banach Spaces
2. Dineen S.- Complex Analysis on Infinite-Dimensional Spaces (1999)
3. Fleming, Jamison - Isometries on Banach Spaces Vol.1 Function Spaces (2003) and Vol.2 Vector-Valued Function Spaces (2008)
4. Michor, Kriegl - The Convenient Setting of Global Analysis (1997): it has two chapters on differential calculus in Banach Spaces IIRC.
5. Omori H.- Infinite-Dimensional Lie Groups (revised,1997): first chapter is about infinite-dimensional calculus.
6. Lang's books Fundamentals of Differential Geometry (2001) and Differential and Riemannian Manifolds (3ed.,1995) have the first chapter dedicated to differential calculus in TVS.
7. Fabian M.- Functional analysis and infinite dimensional geometry (2001)
8. Tsoy-Wo Ma - Classical Analysis on Normed Spaces (1995)


For the sake of completeness, here are some more references that cover the subject quite extensively, that, however, are older than 1995, but not necessarily older than the first edition of Yosida's book (which is from 1980):


9. Frölicher, Kriegl - Linear Spaces and Differentiation Theory (1988)
10. Gil - Norm Estimations for Operator-Valued Functions and Applications (1995)
11. Yamamuro S.- Differential Calculus in Topological Linear Spaces (1974)
12. Zelazko W.- Banach Algebras (1973)
13. Barroso J.- Introduction to Holomorphy (1985)
14. Buoni - Differentiability in Banach Algebras (1974)
15. Coeure G.- Analytic Functions and Manifolds in Infinite-Dimensional Spaces (1974)
16. Dineen S.- Complex Analysis in Locally Convex Spaces (1981)
17. Mujica J.- Complex Analysis in Banach Spaces (1986)


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**EDIT:** To answer OP's last edit. $L^p$-spaces of vector-valued functions are treated for example in Vol.2 of Fleming and Jamison's Isometries on Banach Spaces. More generally, if OP is interested in vector-valued measures, here are some classical references:


18. Bichteler - Integration Theory (with Special Attention to Vector Measures) (1973)
19. Diestel, Uhl - Vector measures (1977)
20. Dinculeanu - Vector Measures (1967)
21. Xia Dao-Xing - Measure and integration theory on infinite-dimensional spaces (1972)


