# 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

If you want to understand well topological vector spaces (TVS), and learn all the basic theorems of functional analysis in their most abstract version, which is usually in the context of TVS and not of normed spaces, I am afraid there is no speedy way. (No royal way to geometry, as the ancient greeks said.)

I would recommend (without any resevations) the book of Walter Rudin, Functional Analysis, which builds the foundations of TVS in a rather elegant way. The books of G.M.H Köthe and H.H. Schaefer I wouldn't recommend for a first reading.

• For topological vector spaces there is also a nice book entitled "Topological Vector Spaces" by L. Narici and E. Beckenstein. – Shailesh Dec 18 '13 at 12:27