Reddit reviews Elements of the Theory of Functions and Functional Analysis (Dover Books on Mathematics)

I'm not going to say I'm great at math or anything, but I can say my power to actually proving things, solving many types of mathematical problems more effectively comes from reading this single fairly short book:

Elements of the Theory of Functions and Functional Analysis by Kolmogorov (the guy that formalized probability!)

It's difficult (especially if you try to solve everything he proposes -- which by the way is fairly essential to get the full experience), but I feel it really made me more mature (and some of the tools presented are actually quite useful in research).

It really shows how important proof is. Needless to say, thoroughly recommended, probably my favorite book ever.

(I heard it uses some uncommon terms due to Russian translation but I didn't have any problems e.g. discussing pure math at stackexchange)

If you've ever wondered "How the hell can someone come up with, and prove formally something such as the Universal approximation theorem", read this book.

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Other than that, I'd say one does not simply know enough linear algebra. Or in general one does not know enough about linear systems (including Markov chains), if you work with any kind of time-varying problems at all.

Fomin and Kolmogorov is a classic.

https://www.amazon.com/Elements-Functions-Functional-Analysis-Mathematics/dp/0486406830

Some of the terminology is out of date but it's a nice exposition.

Try Kolmogorov and Fomin's Elements of the Theory of Functions and Functional Analysis.

PROS:

• clear exposition
  
  
• short
  
  
• good, but not overly demanding exercises
  
  CONS:
  
  
• definitely not a reference text for the field
  
  
• no "applications section" to speak of
  
  
• old, and it shows in the nomenclature