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.