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?


closed as off-topic by Alexandre Eremenko, Yemon Choi, Alex Degtyarev, András Bátkai, Mikhail Katz Feb 2 '16 at 8:42

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    $\begingroup$ Almost any French Calculus textbook: Cartan, Dieudonne, etc. $\endgroup$ – Alexandre Eremenko Feb 1 '16 at 22:08
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    $\begingroup$ What results about Banach space-valued functions? $\endgroup$ – Yemon Choi Feb 1 '16 at 22:55
  • $\begingroup$ Please try to narrow down what you want to know about what kind of Banach-space valued functions. Do you want vector-valued $L^p$? vector-valued holomorphic functions? etc $\endgroup$ – Yemon Choi Feb 1 '16 at 22:57
  • $\begingroup$ @AlexandreEremenko: Could you name a specific one? $\endgroup$ – Jack Feb 1 '16 at 23:11
  • $\begingroup$ @Jack: re your edit about the specific branch of Banach-valued functions, have a look at the edit of my answer. $\endgroup$ – M.G. Feb 1 '16 at 23:42

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):

  1. Frölicher, Kriegl - Linear Spaces and Differentiation Theory (1988)

  2. Gil - Norm Estimations for Operator-Valued Functions and Applications (1995)

  3. Yamamuro S.- Differential Calculus in Topological Linear Spaces (1974)

  4. Zelazko W.- Banach Algebras (1973)

  5. Barroso J.- Introduction to Holomorphy (1985)

  6. Buoni - Differentiability in Banach Algebras (1974)

  7. Coeure G.- Analytic Functions and Manifolds in Infinite-Dimensional Spaces (1974)

  8. Dineen S.- Complex Analysis in Locally Convex Spaces (1981)

  9. Mujica J.- Complex Analysis in Banach Spaces (1986)


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:

  1. Bichteler - Integration Theory (with Special Attention to Vector Measures) (1973)

  2. Diestel, Uhl - Vector measures (1977)

  3. Dinculeanu - Vector Measures (1967)

  4. Xia Dao-Xing - Measure and integration theory on infinite-dimensional spaces (1972)

  • $\begingroup$ How does e.g. Zelazko's book relate to the original question? I admit it has been a long time since I looked at it, but will the OP really find what he is after there? $\endgroup$ – Yemon Choi Feb 1 '16 at 23:08
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    $\begingroup$ @YemonChoi: Chapter III in Zelazko's book is about Analytic Functions in Banach Algebras. Since I don't know which branch of Banach-valued functions the OP is interested in, I have included both the general differential calculus in TVS spaces branch and the holomorphic/analytic branch. $\endgroup$ – M.G. Feb 1 '16 at 23:19
  • $\begingroup$ I still think the OP's question is far too broad and ill-defined; the length of your list would seem to bear that out. $\endgroup$ – Yemon Choi Feb 2 '16 at 0:17
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    $\begingroup$ Well, the first and much bigger part of my answer was posted before the OP specified about $L_p$-spaces (and tbh, at first I did not think of the integration theory at all, but focused on differentiation). Now, with the edits on both sides, it does not seem so bad. Plus, I don't mind posting such an extensive list as I have the references more or less in front of me (I'm interested in noncommutative analysis), so it's no problem at all. Nevertheless, I'll leave the list as it is now, just in case anyone else is looking for an extended list of sources on the topic. $\endgroup$ – M.G. Feb 2 '16 at 0:32
  • $\begingroup$ Fair enough, perhaps the list will be useful for others $\endgroup$ – Yemon Choi Feb 2 '16 at 0:37

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