Learning Aid for Basic Theorems of Topological Vector Spaces in Functional Analysis

manpreet Best Answer 2 years ago

I am self-teaching myself the basics of functional analysis (e.g. topological vector spaces), and frankly I am starting to get a migraine sorting out/organizing in my head all of the details/equivalances/implications of the basic theorems involving the various topologies one places on XX, X∗X∗, and X∗∗X∗∗. In particular, details like the conclusion of the Banach-Alauglu theorem, weakly convex (dense) sets are equivalently strongly convex (dense) sets, closed unit ball in a normed space X is weakly compact if and only if X=X∗∗X=X∗∗ (= meaning isometric isomorphism onto X∗∗X∗∗), when sequential compactness is equivalent to compactness and in which topology/space, linear transformations from X→YX→Y are continuous on the weak topologies if and only if they are continuous on the strong topologies, etc. etc. etc.

Are there a succinct set of notes (or better yet a (large) diagram/table/some-sort-of-graphical-aid) online that clearly organizes the basic theorems regarding the spaces XX, X∗X∗, X∗∗X∗∗ and B(X,Y)B(X,Y)(distinguishing further the type of space XX initially is: general TVS, locally convex, normed, Banach, Hilbert, etc.) and the various topologies one places on them (strong, weak, weak-*, strong operator, weak operator, etc.)

I realize that this question is very broad, and I'm sure this is just one of those things that takes a lot of time and dedication in order to get a strong feeling for, but I wanted to ask nonetheless since I am sure there are other people in the same situation.