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

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 $X$, $X^{*}$, and $X^{**}$. 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 $X=X^{**}$ (= meaning isometric isomorphism onto $X^{**}$), when sequential compactness is equivalent to compactness and in which topology/space, linear transformations from $X\to Y$ 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 $X$, $X^{*}$, $X^{**}$ and $B(X,Y)$ (distinguishing further the type of space $X$ 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.

• +1 for the OP. I also doing a self study on topological vector spaces.:) – Juniven Mar 13 '13 at 8:42
• Perhaps it might be best to sort out these things for normed linear spaces before attacking general TVS's. – Prahlad Vaidyanathan Dec 14 '13 at 5:06
• The more you study TVS, the worse it gets. I did my Ph.D. in Banach spaces (a special case of TVS), and the subject is full of difficult counterexamples to otherwise plausible conjectures. There are not many unifying ideas. – Stephen Montgomery-Smith Dec 19 '13 at 13:39
• I can also suggest the book by Koether, which I think is called "Topological Vector Spaces." It was translated into English by my Ph.D. adviser. It is more encyclopedic than an easy read, so probably a good book to have around rather than try to read like a novel. – Stephen Montgomery-Smith Dec 19 '13 at 13:41
• The Wikipedia page 'Operator topologies' has a section Which topology should I use? that seems kind of along these lines. – Ruvi Lecamwasam Apr 23 '17 at 23:10