Functional Analysis & Others

[intwo]

I'm looking for a rigorous introduction to functional analysis in the style of Apostol. I've looked at Introductory Functional Analysis with Applications by Kreyszig, but I find it slightly too conversational. I know that Rudin has a Functional Analysis book, but it seems to be out of print and my library does not have a copy to view.

I'm also looking for textbook on Vector Analysis. The treatment in Apostol's Calculus seemed insufficient and was nonexistent in his Mathematical Analysis.

And are there any books that rigorously cover coordinate transformations? I learned the basics of spherical and cylindrical coordinates in multivariable calculus and used them frequently in physics courses, but I've never seen them treated theoretically. Apostol's books contained Jacobians, but did not treat problems in other coordinates in depth. I understand that coordinate systems might just be a tool for applications and may not exist in a theoretical context, but I thought I'd ask anyways.

Thanks!

 

[micromass]

There are many approaches to functional analysis. Kreyszig is a good book, but it asumes no topology or measure theory, and can therefore go not so deep.

I really like "Analysis now" by Pedersen. It's a really neat book that contains most material on functional analysis.

A somewhat easier book is "introduction to hilbert spaces with applications" by Debnath and Mikusinski. The great thing about this book is that it covers a LOT of applications.

The book "Functional analysis" by Lax is also a piece of work. It even contains many historical notes. It's quite depressing though as it talks about all those great mathematicians who died in the holocaust. But the book is certainly very good.

I suggest you check out these three and take what you like most!

 

[morphism]

What topics do you want to learn about?

From my experience, the functional analysis book closest in style to Apostol is Conway's A Course in Functional Analysis (Springer GTM).

I also enjoyed the book by Pedersen, which micromass mentioned, but be warned: he leaves lots of important (and not so easy) results to the exercises, without really pointing out that they're important (and in some cases without indicating that they're named theorems).

In any case, both Conway and Pedersen have plenty of really nice and challenging exercises, so they're good books to work through.

 

[intwo]

Thank you for your responses, micromass and morphism.

What topics do you want to learn about?

I've taken upper level physics courses that frequently used mathematics without explanation and I feel as though I've accumulated a superficial appreciation of some of the topics. In quantum mechanics we briefly mentioned Hilbert spaces and freely used Fourier transforms and other "techniques" to solve problems, but I never felt comfortable applying the mathematics.

I am now attempting to reconcile my mathematical qualms by working through theoretical textbooks. Mathematics makes more sense (to me) in terms of definitions and theorems than in an "apply this to that" context. I actually had difficulty remembering basic concepts in calculus, linear algebra, and differential equations until I relearned them in a theorem-proof format.

I apologize for the tangent, but I thought some background information could be beneficial. I want to learn functional analysis so that I can apply it in physics, but I also appreciate the pure mathematics. I prefer a book that contains most of the functional analysis used in physics, but it need not contain any applications - I can get the applications from my physics books.

I've skimmed through your suggestions on Amazon and they seem fitting, but I have not taken a course on topology or a formal course on measure theory besides the minor acknowledgments in real analysis. Perhaps a more appropriate question would pertain to measure theory - what is a rigorous book on measure theory? Topology by Munkres seems to be the standard topology textbook so I'll probably pick that one up in the future unless anyone has any other suggestions.

Thanks again for your responses!

 

[micromass]

Ah, you never learned about topology and measures before. That makes it somewhat more annoying. I fear that Kreszig is the only functional analysis book out there that you can read at the moment.

Here are 3 other books:
- https://www.amazon.com/dp/0120502577/?tag=pfamazon01-20 deals with topology, measures and functional analysis (a bit).You might find it a bit challenging though.

- https://www.amazon.com/dp/0763717088/?tag=pfamazon01-20 is a fun and not so hard intro to measure theory.

- https://www.amazon.com/dp/0201002884/?tag=pfamazon01-20 is a book that deals with so much. It's quite rigorous though.