Need a Linear Algebra Book for Quantum Mechanics?

[fluidistic]

I've had a proof based linear algebra course as a freshman, where I learned that the spectrum of an operator was the set of the eigenvalues of that operator. Now in quantum mechanics I learned that this isn't true and that the spectrum of an operator can contain infinitely more numbers.
Also in my course I've never learned anything about vector spaces of infinite dimension.
I'm getting lost with the linear algebra part of QM. Could you please recommend me some book(s) that deals with linear algebra (better if it's aiming at physicists) with vector spaces of infinite dimension?
Thank you very much.

 

[WannabeNewton]

You're asking for books on functional analysis. Without making any assumptions about your knowledge of topology and real analysis, I can recommend this book: https://www.amazon.com/dp/0471504599/?tag=pfamazon01-20. It is light on topology and measure theory but it has the benefit of covering all the necessary material needed on metric spaces and LA. If you perhaps want a more concise summary of the functional analysis used in QM, take a look at chapter 1 of Ballentine.

EDIT: btw I remember you told me that your professor doesn't use bra-ket notation. Chapter 1 of Ballentine has the added benefit of showing you why it works (i.e. Riesz representation theorem) and then he proves a good number of theorems whilst using the notation so it should give you good exposure to the notation if that is still of importance to you.

 

[fluidistic]

You're asking for books on functional analysis. Without making any assumptions about your knowledge of topology and real analysis, I can recommend this book: https://www.amazon.com/dp/0471504599/?tag=pfamazon01-20. It is light on topology and measure theory but it has the benefit of covering all the necessary material needed on metric spaces and LA. If you perhaps want a more concise summary of the functional analysis used in QM, take a look at chapter 1 of Ballentine.

EDIT: btw I remember you told me that your professor doesn't use bra-ket notation. Chapter 1 of Ballentine has the added benefit of showing you why it works (i.e. Riesz representation theorem) and then he proves a good number of theorems whilst using the notation so it should give you good exposure to the notation if that is still of importance to you.

Ok thanks a lot, I didn't even know about functional analysis. I've never taken a topology course, nor real analysis.
Unfortunately the library of my university lacks both books. If I buy from amazon (assuming I've no problem with the prices, which isn't a given), at best I'll have the book in about 1.5 month, basically too late for my course.
At least now I know where to look at: functional analysis.
About the bra ket notation you're right, my prof. doesn't use it in public. I've borrowed Sakurai and Messiah's books yesterday though in order to learn it.

 

[micromass]

Another good book is Prugovecki: https://www.amazon.com/dp/0486453278/?tag=pfamazon01-20

I do think it's more a math books than a physics book, though. But it presents all the math is a nice and rigorous way.

 

[fluidistic]

Another good book is Prugovecki: https://www.amazon.com/dp/0486453278/?tag=pfamazon01-20

I do think it's more a math books than a physics book, though. But it presents all the math is a nice and rigorous way.

Ok thank you very much. Apparently they have it at the library... going for it on Tuesday.

 

[dx]

I would also recommend this book for an introduction to functional analysis for quantum mechanics:

https://www.amazon.com/Mathematical-Physics-Chicago-Lectures/dp/0226288625/ref=sr_1_1?s=books&ie=UTF8&qid=1366483323&sr=1-1&keywords=mathematical+physics+geroch

See chapters 48 to 56 in particular.