What's the Next Step in Math After Real Analysis?

[the_kid]

I'm looking for a little advice on where I should direct my mathematics studies this summer. My last math course was a rigorous real analysis course covering the first eight chapters of Rudin's book. What would be the next logical topic (in analysis) to study? I'm thinking I should do some work with Lebesgue integration and measure; could someone recommend a book for this? When will I be prepared for functional analysis? Any advice is much appreciated.

 

[micromass]

If you want to, you can do functional analysis right now. The book by Kreyszig doesn't need measure theory or topology but does get quite far. Of course, not treating measure theory has its negative sides as the L^p spaces aren't treated well enough.

Perhaps the best thing to do now is to go a bit further in real analysis and try to get a book with a view on functional analysis. An excellent book for this is "Principles of Real analysis" by Aliprantis and Berkinshaw. It develops topology, measure theory and the theory of Hilbert spaces.

Other good books are the real analysis book by Carothers.

 

[mathwonk]

i liked spectral theory, by edgar lorch, as an undergrad. but you need some complex analysis also.

 

[the_kid]

If you want to, you can do functional analysis right now. The book by Kreyszig doesn't need measure theory or topology but does get quite far. Of course, not treating measure theory has its negative sides as the L^p spaces aren't treated well enough.

Perhaps the best thing to do now is to go a bit further in real analysis and try to get a book with a view on functional analysis. An excellent book for this is "Principles of Real analysis" by Aliprantis and Berkinshaw. It develops topology, measure theory and the theory of Hilbert spaces.

Other good books are the real analysis book by Carothers.

Thanks for the reply! Another quick question, if you don't mind: what gaps do I need to fill into be able to do functional analysis "the right way"? I.e. what would you call prerequisite material.

 

[micromass]

Thanks for the reply! Another quick question, if you don't mind: what gaps do I need to fill into be able to do functional analysis "the right way"? I.e. what would you call prerequisite material.

- Measure theory
- Topology
- Linear algebra
- Real Analysis
- A bit complex analysis

The real analysis book by Serge Lang is really, really good in covering prereqs. It is certainly worth a read, but it's no easy book.

 

[the_kid]

- Measure theory
- Topology
- Linear algebra
- Real Analysis
- A bit complex analysis

The real analysis book by Serge Lang is really, really good in covering prereqs. It is certainly worth a read, but it's no easy book.

Thanks for the reply, micromass. I'm looking to take a course that is described below:

"Measure Theory and Integration--
Construction and limit theorems for measures and integrals on general spaces; product measures; Lp spaces; integral representation of linear functionals."

As I mentioned, I'm familiar with the first eight chapters of Rudin. Would this be sufficient preparation for such a course or is there any material I need to bridge the gap? Thanks for your help!

 

[micromass]

Yeah, a basic knowledge of real analysis (such as Rudin) should absolutely be sufficient to get through the course alive. I think you're good to go.

Basically, all you need to know is a bit topology (limits, continuity, compactness, Heine-Borel, etc.).

 

[the_kid]

Awesome--thanks! On a related note, what book would you recommend for material covered in that course?

 

[micromass]

Check the books in my post 2. The books by Aliprantis and Carothers are very good!