Topological vector space textbook with enough applications (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?
 A: *

*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
A: 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,...
A: 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. 
A: 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.
A: Functional analysis book by Kreyszig.
