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I'm a mathematics student in abstract algebra and algebraic geometry. Most of my books cover a great deal of category theory and it is an essential tool in understanding these two subjects.

Recently, I started taking some functional analysis courses and I discovered that there is almost no category theory done in these courses. But since most of the spaces studied in functional analysis are objects in categories (e.g. the normed spaces form a category), I find it rather strange that the books leave the category theory out.

Is there a reason for this?

  • 4
    I suppose there are a lot of reasons they don't use category theory, and hence don't need to cover it. Do you see some areas of analysis where it would be of use?2012-07-10
  • 1
    @gary : A wild guess at why they don't learn it is because even though many objects in analysis can be viewed from a categorical point of view, perhaps there is no use of using this for them. But that's just a wild guess ; I am always careful when I speak about categories, they tend to say powerful things..2012-07-10
  • 2
    Category theory provides an interesting perspective when looking at pretty much any part of mathematics. However, if it does not find any direct application (yet), there's no wonder they're not introduced in textbooks. Neither is formal logic, or even slightly more in-depth set theory (which is a little more directly related to functional analysis, I believe).2012-07-10
  • 2
    Category theory is used in Bergh-Löfström's book on interpolation spaces.2012-07-10
  • 0
    There are some analysis references that do, in fact, apply category theory. For instance, see Schecter's Handbook of Analysis and its Foundations2012-07-10
  • 9
    I don't have a good answer to your question, but this MO question might be of interest to you: http://mathoverflow.net/questions/38752/analysis-from-a-categorical-perspective2012-07-10
  • 10
    You may wish to consult Helemskii's *[Lectures and Exercises on Functional Analysis](http://books.google.com/books?id=wjzZCLzx6hUC)*.2012-07-10
  • 1
    In my extremely limited experience of category theory, I have found that it provides an interesting unifying perspective after the fact. The level of abstraction is too high for me to fruitfully use category theory.2012-07-10
  • 5
    Another somewhat related MO question: http://mathoverflow.net/questions/22359/why-havent-certain-well-researched-classes-of-mathematical-object-been-framed-by2012-07-10
  • 5
    Hummm, and here I had always thought analysts did quite a bit with [category notions](http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?handle=euclid.rae/1337001353&view=body&content-type=pdf_1) :)2012-07-10
  • 0
    [Related question](http://math.stackexchange.com/questions/90981/is-category-theory-useful-in-higher-level-analysis) which was [cross-posted to MO](http://mathoverflow.net/questions/83363/is-there-a-nice-application-of-category-theory-to-functional-complex-harmonic-an).2014-10-16

6 Answers 6