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(Sorry for my bad English.)

For "applications", I mean applications in math, not real-life.

There are many textbooks about topological vector space, for example, GTM269 by Osborne, Modern Methods in Topological Vector Spaces by ALBERT WILANSKY, etc.

Most textbooks make many definitions, and proved many theorem of their properties, but with very few application.

For example, in GTM269 preface, the author says "Although this book is oriented toward applications, the beauty of the subject may appeal to you."

But most theorems in this book really don't have any application (in book).

So, are there some topological vector space textbook (about generally topological vector space, Frechet space, locally convex space or this kind of spaces. Not Banach space or Hilbert space), which most theorems have applications?

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    $\begingroup$ @AlexM.: Meta is that way -->. But topological vector spaces aren't a typical undergraduate topic in my universe, and all the books I recognize in the answers are what I would unambiguously call graduate texts. The Wilansky book is marketed "for advanced undergraduate and beginning graduate students", but I've found that such tags are often wishful thinking from publishers hoping to sell to a larger audience. $\endgroup$
    – Nate Eldredge
    Commented Feb 8, 2018 at 14:28
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    $\begingroup$ related question: mathoverflow.net/questions/259834/… $\endgroup$
    – Abdelmalek Abdesselam
    Commented Feb 8, 2018 at 16:04
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    $\begingroup$ But distribution is not the only LF space... $\endgroup$
    – QiRenrui
    Commented Feb 8, 2018 at 16:30
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    $\begingroup$ @QiRenrui: make up your mind. Do you want applications or not? If you do, most of your examples will come from distributions. Of course, one has to be flexible in what one means by distribution and include for instance: distributions on local fields and adeles, currents, distributions on noncompact Lie groups a la Harish-Chandra, etc. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Feb 8, 2018 at 17:05
  • $\begingroup$ What, for you, counts as an application? Would e.g. number theory be an application? Differential geometry? Or do you want things like the design of turbojet engines? $\endgroup$
    – Yemon Choi
    Commented Feb 9, 2018 at 4:03

5 Answers 5

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  • Hormander: The Analysis of Linear Differential operators I-IV:

  • Reed-Simon: Methods of Mathematical Physics I-IV

  • Treves: Topological Vector spaces, Distributions and Kernels

  • Taylor: Partial Differential Equations I-III

  • Taylor: Pseudodifferential Operators and Nonlinear PDEs

  • Gelfand-Shilov: Generalized Functions I-V

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Walter Rudin's Functional Analysis has many applications e.g., vector measures, generalized Stone-Weierstraß theorems, interpolation results, Fourier analysis, distribution theory, elliptic partial differential equations, prime number theorem, ergodic theorems,...

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Topological vector spaces, other than Banach spaces with most applications are Frechet spaces. The primary sources arei: L. Schwartz, Theorie des distributions, 1966, and I. Gelfand, G. Shilov, Generalized functions, vol. 1 (the other volumes contain applications). And there are hundreds of secondary sources.

EDIT. Let me add a book-size survey on applications of Frechet manifolds:

MR0656198 Hamilton, Richard S. The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65–222.

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I would recommend you the book by Yu.I.Lyubich (an unfavourable Zentralblatt review is here, see comments below). It's a good introduction to functional analysis for people who are interested in applications.

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  • $\begingroup$ The linked zbmath review is unfavorable - what was your opinion on the features that concerned the reviewer? $\endgroup$
    – Nate Eldredge
    Commented Feb 8, 2018 at 14:25
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    $\begingroup$ Nate, excuse me, I read this review only after your comment. I must say, I did not understand Appell's reproaches to Lyubich. Lyubich builds his exposition as a historical review explaining which problems lead to different ideas of functional analysis. That is why for example topological vector spaces appear only in the middle of his book. Of course this style is original and the result can't be treated as the only possible source for studying functional analysis. But for people who are interested in applications and in the origin of the ideas it's a very useful text. $\endgroup$
    – Sergei Akbarov
    Commented Feb 8, 2018 at 15:36
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Functional analysis book by Kreyszig.

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