Last summer I learned some quantum mechanics and thought it was pretty interesting. Some of the theoretical parts use a lot of functional analysis, so I thought that functional analysis would be a fun class to take. After all, its name contains the word "fun", so how bad could it be?
While some of the topics in the class were fairly interesting, some other topics were boring and complicated and pointless. The homework problems were extremely difficult, which is not bad by itself. But, more importantly, upon reading the problem I would rarely see why this is a problem that needs solving. There would be so many steps, each one being a small result that I don't care about, which when taken together prove a larger result that I also don't care about. So, since I wasn't motivated to do the homework, halfway through the semester I decided to audit the class.
I did learn some cool stuff, though the specifics often eluded my understanding. I learned the basics of Banach spaces and Hilbert spaces. I also learned about bounded linear transformations; the term "bounded" here means that there is an upper bound on the growth of a vector when it is put through the transformation. In finite-dimensional vector spaces (that is, Rn), all linear transformations are bounded; however, in infinite dimensions this is not always the case, and often it is necessary to consider only the bounded transformations.
Another cool thing is about eigenvalues. In finite dimensions, all the values of λ for which A–λI is non-invertible are eigenvalues of A; you have probably used this fact many times if you have ever taken linear algebra. With infinite-dimensional linear transformations, there may be some values of λ for which A–λI is non-invertible and yet λ is not an eigenvalue. The former set is called the spectrum, and the spectrum always contains the eigenvalues—but not everything in the spectrum is an eigenvalue. The professor showed us an example of a transformation whose spectrum includes the real numbers in the interval (–2, 0) but which has no eigenvalues at all.
There are several phenomena like these: two distinct concepts which always happen to coincide in finite dimensions don't always coincide in infinite dimensions. I'm glad that I learned what I did; it gave me a new perspective on some parts of mathematics. I also learned something about myself: analysis isn't really my cup of tea.
Another cool thing is about eigenvalues. In finite dimensions, all the values of λ for which A–λI is non-invertible are eigenvalues of A; you have probably used this fact many times if you have ever taken linear algebra. With infinite-dimensional linear transformations, there may be some values of λ for which A–λI is non-invertible and yet λ is not an eigenvalue. The former set is called the spectrum, and the spectrum always contains the eigenvalues—but not everything in the spectrum is an eigenvalue. The professor showed us an example of a transformation whose spectrum includes the real numbers in the interval (–2, 0) but which has no eigenvalues at all.
There are several phenomena like these: two distinct concepts which always happen to coincide in finite dimensions don't always coincide in infinite dimensions. I'm glad that I learned what I did; it gave me a new perspective on some parts of mathematics. I also learned something about myself: analysis isn't really my cup of tea.
1 comment:
"The homework problems were extremely difficult, which is not bad by itself. But, more importantly, upon reading the problem I would rarely see why this is a problem that needs solving. There would be so many steps, each one being a small result that I don't care about, which when taken together prove a larger result that I also don't care about."
Interesting, that's exactly how I feel when reading papers from certain subfields of linguistics. Though I love thinking about language and I think that figuring out how language works is an extremely important human endeavor (obviously, otherwise I wouldn't be majoring in it), some linguists just seem to spend ages researching phenomena that are not very important and explaining them with extremely complicated theories that have no relevance to anything. I guess no field is 100% exciting.
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