# Lebesgue space Q & suggested readings

1. Oct 23, 2014

### tigigi

I wonder if anyone can suggest any short reading about basic understanding of L space or L2 space (not wikipedia) I know nothing about it, googled around, but doesn't seem to find things helping me solve some problems.

First, there's this question: For what range of μ is |x|^μ in L2(-1,1)
I thought the def for L2( [a,b]) = {x:(a,b) -> c | x2 is finite}, x2 = integrate x(t)^2 over t from a to b
but it's confusing, where's t? Also, it doesn't seem to be able to solve this problem

Any suggestion? Thanks.

2. Oct 23, 2014

### homeomorphic

Actually, that is how you solve the problem. You just seem not to have pursued it to the finish line. Lack of knowledge of L^2 doesn't seem to be the issue in this case.

The definition is actually that the square is Lebesgue integrable, which can take you into the deep waters of measure theory, but Riemann integrable implies Lebesgue integrable, which is good enough for your problem. If you want to learn measure theory, I like A Radical Approach to Lebesgue's Theory of integration, but that assumes some comfort level with epsilons and deltas from introductory real analysis. If you really want to know what L^2 is, you have to come to terms with that stuff. I don't know of any really short reading.

3. Oct 23, 2014

### Staff: Mentor

Here are two books I have that will explain all, and much more besides, if you have calculus knowledge equivalent to US Calculus BC, IB Math HL, first year university calculus etc:
http://matrixeditions.com/UnifiedApproach4th.html
http://matrixeditions.com/FunctionalAnalysisVol1.html

The reviews are correct - simply superb.

Read the Calculus and Linear Algebra book (it explains Lebesgue integration, Fubini's theorem etc), then Functional Analysis.

I was very fortunate in my undergrad degree studying two subjects that covered that material, but many degrees don't. Which IMHO is a pity. As an aside my old school no longer even teaches them nor even requires people to do their epsilonics - which of course I believe is a backward step. But to be fair students hated analysis and virtually no-one took them in later years.

That said if your primary interest is QM such technicalities aren't really required to start with - you can learn them later if its mathematical foundations appeals. And if you do that then Rigged Hilbert Spaces, rather than just L2 Hilbert spaces will be required:
http://physics.lamar.edu/rafa/webdis.pdf [Broken]

The two books I mentioned would provide sufficient background to understand the above paper.

A word of warning though - it will take a while :):):):):):):):):)

Of course its worth it.

Thanks
Bill

Last edited by a moderator: May 7, 2017