How to learn functional analysis

[wdjhit]

I major in physics, but I'm also very interested in mathematics, especially analysis. Until now, I have taken mathematical analysis and real analysis. Now, I want to learn functional analysis by myself,
and my teacher adviced me to read topology first. But I found it difficult to understand and may take too much time. So, do I really need to know topology before I study functional analysis? Can you give me some advice?

 

[micromass]

I major in physics, but I'm also very interested in mathematics, especially analysis. Until now, I have taken mathematical analysis and real analysis. Now, I want to learn functional analysis by myself,
and my teacher adviced me to read topology first. But I found it difficult to understand and may take too much time. So, do I really need to know topology before I study functional analysis? Can you give me some advice?

No! You don't need to know topology for functional analysis (although it cerainly helps). A very good book that introduces functional analysis at a rather basic level (meaning: no prereq knowledge of topology and measure theory) is Kreyszig: https://www.amazon.com/dp/0471504599/?tag=pfamazon01-20 The book goes rather far without assuming too much. Of course, with only a basic prereq knowledge assumed, it can't do everything in the "best" possible way. For example, the $L^2$ spaces are not treated in the most intuitive way, since it requires measure theory.

Another good book is Reed & Simon: https://www.amazon.com/dp/0125850506/?tag=pfamazon01-20 However, it is rather terse, and I wouldn't recommend it to an undergrad without much experience in pure math. He doesn't assume topology and measure theory, but he goes over it far too quickly to be of any use to an undergrad. It's one of my favorite books though.

More advanced books definitely need a good and solid knowledge of topology and measure theory.

If you're interested in PDE, then this book is good: https://www.amazon.com/dp/0387709134/?tag=pfamazon01-20 Although it might be too advanced for an undergrad.

If you're interested in QM, then this book is excellent: https://www.amazon.com/dp/0486453278/?tag=pfamazon01-20 It only deals with Hilbert spaces though, and not with Banach spaces. But he does go a long way without many prereqs.

 

[wdjhit]

Thanks very much! Actually I have the book written by Kreyszig. It is simple, and clear, and also because of this I'm worried whether the book can offer me all the materials that a functional class covers. Anyway, I can have a look at the other books you recommended and finally choose one for further studying. Many thanks!

 

[micromass]

Thanks very much! Actually I have the book written by Kreyszig. It is simple, and clear, and also because of this I'm worried whether the book can offer me all the materials that a functional class covers. Anyway, I can have a look at the other books you recommended and finally choose one for further studying. Many thanks!

Yes, it covers all that a first class on a topic covers. You can of course go way beyond this. It depedends on what your goal is and why you study it.

 

[WannabeNewton]

If you've taken real analysis, why not learn point-set topology anyways? It isn't difficult if you have the right textbook (i.e. don't start off with Willard!) and imo easier than functional analysis.