# Best book on general topology for functional analysis

I study functional analysis. From time to time I find that the basics of many concepts lie on the concepts of topology.

I am familiar with basic concepts of topology. When I go deeper then wild things start.

I am not interested in topology for the sake of topology. I need it for a deep understanding of functional analysis.

Question 1: Is there any good book on topology for my purpose?

Question 2: If I really want to be good in functional analysis do I need study the whole topology or I can be satisfied by the standard topology?

Question 3: My question shows my lack of understanding and immaturity in mathematics. I should not be distracted from the main curriculum. Yes or no?

(My area of interest is harmonic analisys and operator theory. I am newbie).

• Topology is a wide area. I "everything" you need to know is contained in the first part of Munkres Topology. It is a standard reference, and a good book. As I am not interested in functional analysis, there might be better suited books, and I can not tell how much of general topology you should really know. Aug 15, 2021 at 16:42
• See the following PDF: mat.univie.ac.at/~mike/teaching/ss16/general_topology.pdf Aug 16, 2021 at 1:03
• Robert Ash's Real Analysis and Probability is a book of a kind, it has essentially three parts. Part 1 is the topology appendix where he derives most properties needed en the first four chapters (which would be Part 2) and these chapters start with measure theory (ch1-2) and then they continue with the theory of linear forms in Frechet spaces (ch3-4). (After that, he goes on to study probability theory, so it's kinda funny that after chapter 4 where he proves Baire's theorem and it's consequences, in ch5 he defines Bernoulli random variables, typically studied without regard of analysis). Aug 16, 2021 at 2:19

## 1 Answer

1. Any standard book on point-set topology would work. Munkres is the canonical recommendation, and is good.
2. What do you mean the 'whole topology' and the 'standard topology'? Regardless, for functional analysis you'll probably just need regular point-set topology, nothing algebraic. For real analysis (i.e., what you studied before functional), you probably needed very little topology--just some ideas about open and closed sets. But for functional analysis, you want to understand very well the product topology and Tychonoff's theorem, so that the weak and weak* topologies make sense.
3. I'm not entirely sure how to parse this question. I will say that it's a bad idea to study functional analysis if you don't understand very well topology and real analysis (including measure theory) very well already. Particularly, you should be able to do the problems in, say, the first half of Rudin's Real and Complex Analysis. (Without a solid background in real analysis, functional analysis will make little sense and you won't have the right examples. If you go in without understanding $$L^p$$ spaces and how to use them, then a lot of the motivation for functional analysis will be lost. Rudin is nice for this since his book includes the beginnings of functional analysis--basically everything you can do without getting into hard material: Analyzing basic properties of Banach spaces with attention paid to $$L^p$$, and using Hilbert spaces to solve certain optimization problems and develop the theory of Fourier series. Going for an abstract treatment of functional analysis without knowing these concrete motivations is a very big mistake.) Also, know finite-dimensional linear algebra down cold. You should be able to prove the spectral theorem in finite-dimensions, know about canonical forms of matrices, etc. like the back of your hand. Linear algebra is useful in all branches of mathematics, and a large part of functional analysis is trying to extend the tools of finite-dimensional linear algebra to make sense in infinite-dimensions. If you don't understand very well the finite-dimensional case, a lot of the motivation for functional analysis will be lost.
• This is a good answer. After looking through Munkres text, it might also be helpful to have Kelley’s General Topology available as a reference. There, he has written the text specifically geared towards analysts (though it isn’t particularly “introductory” so I recommend Munkres first) Aug 15, 2021 at 17:11
• I have read the first chapter in Munkers text book. I will continue. Thanks! Aug 15, 2021 at 17:16