Suggestions for math methods book

[onoturtle]

I'm sorry for yet another one of these threads, but I'm curious if this will lead me to a better reading for me.

I want to beef up my math background so I decided to take a look Hassani's Mathematical Physics and I planned on reading the parts on vector spaces (finite/infinite), complex analysis and group theory. Skipping the ODE/PDE stuff since I'm not that interested in it. However, working through the second chapter, I am bothered by the inconsistency of the author giving or not giving concrete examples. E.g. while the author did give examples of vector spaces, he doesn't for concepts I'm not familiar with: tensor product and complex structure. While I did look up tensor product and Dirac bra-ket notation on Wikipedia that helped clarify things (e.g. outer product as an example of tensor product), it bothers me that the author sometimes skips examples (not even in the exercises!) and I feel like this does not bode well when I reach more advanced chapters mostly filled with concepts I don't know.

I've seen Boas' book suggested numerous times in other threads and I'll probably take a look at the related chapters for complex analysis before Hassani's since that's a new subject for me. But otherwise, Boas' book doesn't seem too interesting to me and it seems like I may find a book that's between Boas' and Hassani's in level of abstractness more enjoyable to read.

To give you a better sense of what level I may be prepared for, I'm a computer scientist that took undergraduate linear and abstract algebra years ago (I don't know about group representation, which is one reason for reading Hassani's) and informally picked up some analysis from studying machine learning. Complex analysis will be a new topic for me.

 

[Geofleur]

Hassani may not be the best book for you, but it seems worth saying how I use it, because the same idea is helpful when reading lots of things, and it might help you with Hassani's book.

When reading Hassani, I make up my own examples (or get them from other supplementary readings) and write them in the margins. They can be simple examples. For instance, regarding complexification using a complex structure $\mathbf{J}$, consider $\mathbb{C}$ as a vector space over $\mathbb{R}$ with basis $\{1,i\}$. If we set $\mathbf{J} = i$, then it's easy to see that $\mathbf{J}$ meets the requirements for a complex structure, that is, $\mathbf{J}^2 = -\mathbf{1}$ and $\langle \mathbf{J} a | \mathbf{J} b \rangle = \langle a | b \rangle$ for all $| a \rangle$, $| b \rangle$ in $\mathbb{C}$ (here I have set $\mathbf{1} = 1$). Now consider the subspace of $\mathbb{C}$ spanned by $1$. Define multiplication by a complex number via $(a + ib)(1) \equiv (a\mathbf{1} + b\mathbf{J})(1)$. Now, by setting $a$ and $b$ appropriately, we can reach any element of $\mathbb{C}$, and we have turned $\mathbb{C}$ over $\mathbb{R}$ into $\mathbb{C}$ over $\mathbb{C}$, a space of half the dimension.

 

[onoturtle]

Peeking ahead, Chapter 5 of Hassani's is on matrices and it appears, at a glance, to be a concrete chapter that will provide examples of the concepts in the preceding chapters. E.g. I did spot an example of a complex structure and the author works out the matrix for it given whatever basis he is using for the vector space. Referencing this chapter hopefully will help my understanding Part I of the text, so I plan to continue on with Hassani.

One reason I'm reading Hassani is to better understand Hilbert spaces (Part II of text) since it is used in one area of machine learning (kernel methods) that I'm interested in. Typical intro ML texts gloss over the details and texts specific to kernel methods assumes math background I'm currently trying to attain. From my cursory glance on how to learn about Hilbert spaces, it seems like it is a topic in graduate analysis texts. Since my knowledge of undergrad analysis isn't that great, I've presumed such a grad math text would be too difficult for me at this point. So hopefully Hassani's treatment works out fine for me to get into the ML literature I want to read.

 

[Geofleur]

Yes, I hadn't thought to mention it but, because matrices are concrete representations of linear operators and vectors, the chapter on matrices does provide some good examples of earlier material!

I like his coverage of Hilbert spaces, and a lot of the concepts from finite dimensional spaces carry over.