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I am a master student and working in PDE area. I am trying to gain deep understanding of some of the concepts in functional analysis which are common tools in PDE research, such as weak*-topology, weak-topology, distributions theory, to name a few.

There exist many books which are appropriate for beginners in PDE. For example, "Functional Analysis, Sobolev Spaces and Partial differential Equations" by Brezis or "Functional Analysis and its Applications" by Peter Lax are good references for this aim. But in all these books I think the authors try to neglect the details of concepts like weak topology, weak convergence and every concept from the functional analysis which needs to be precisely understood. On the other hand, "Functional Analysis" by Rudin is albeit too rigorous book in this context. So it seems to me that I must firstly study the General theory of Topological Vector Spaces from Rudin's book or any other similar book in order to understand fully the concepts, and then start to read books by Lax or Brezis.

I would like to hear yours advises or comments.

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    $\begingroup$ The concept of weak topology/convergence is very well discussed in Chap.3 of Brezis' book. This will cover 99% of the situations you will encounter when investigating pde-s. In fact I would recommend that you first read Brezis before you read more sophisticated discussions of spaces in duality. In particularly, I have not seen anywhere discussed so efficiently the direct method in the calculus of variations (Corollary 3.23 in Brezis). $\endgroup$ – Liviu Nicolaescu Apr 25 '16 at 10:48
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I was in a similar situation many years ago during a seminar in graduate school. At that time I wrote up several notes. Within these notes (a link is given below) are many (hopefully useful) details surrounding the use of weak and weak-* convergence in PDEs, albeit the work only illustrates one example of a Galerkin method for obtaining a local weak solution to a particular problem. See these NOTES.

--A good reference for learning more on distributions is Introduction to The Theory of Distributions, Friedlander and Joshi (also the appendix of this text contains a decent brief treatment of topological vector spaces).

--I found the following nice for dealing with Galerkin method, spectral theory, weak/weak-* convergence:

a. Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Roger Temam
b. Infinite-Dimensional Dynamical Systems, James Robinson
c. An Introduction to Semiflows, Milani and Koksch (disclaimer: Milani was my PhD adviser)
d. Attractors of Evolution Equations, Babin and Vishik

--In many instances one can deal with mild solutions coming from semigroup theory (and in some instances these mild solutions are the same as weak solutions---this depends on the adjoint of the linear operator associated with the abstract Cauchy problem). I found the following useful:

a. Applied Semigroups and Evolution Equations, Belleni-Morante
b. Nonlinear Evolution Equations, Zheng
c. Semigroups of Linear Operators and Applications to Partial Differential Equations, Pazy
d. Semigroups of Linear Operators, Goldstein

Cheers,

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