Uniform Continuity over Compact Sets

[@JoelRosenfeld]

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[@siddhikantmishra]

Thanks for this! I am preparing for my PhD qualifying exams from Rudin. These videos are going a long way in helping me understand the text better. Keep 'em coming!

[@JoelRosenfeld]

I’m glad to hear it! Good luck!

[@mmariokart231]

It’s gold, these analysis/rudin proofs are the best vids on KZbin!!

[@JoelRosenfeld]

Thank you so much! I’m glad you like them!

[@zoedesvl4131]

How the term "topological space" was finally coined? I think I get at least 70% right: 1. before the 20th century, mathematicians kinda had some vague idea about topology, and the word topology was explicitly stated in the 1870s. 2. Poincaré spotted the potential and made mathematicians aware of this "thing". 3. Fréchet gave rise to metric space, which I believe was quite new. 4. Hausdorff, when writing his Set Theory book, thought the idea of metric space was good so he tried to generalise that, omitting the consideration of a real function (i.e. metric). Mathematicians were cool with that, but later they found that Hausdorff's definition was too strong (for two points there are two open sets...) so later it was dropped (perhaps by Bourbaki?). But that condition was considered Hausdorff condition for obvious reasons thereafter. 5. Category theory can be originated from algebraic topology. Certainly Fréchet's contribution shouldn't be underappreciated.

[@Cxlver]

i love the content. thank you for making the videos

[@JoelRosenfeld]

I’m glad you like it!

[@ozymandias4488]

Continuity is an incredible concept, good video! I would like to hear your opinion on this or advices. I'm a physics student and starting with analysis on my own to later go for (hopefully) complex and measure theory to have a good sense on what Im doing, the thing is that I don't want to be an expert on the subject and won't take a more advanced notion of analysis, I Just want a good grasp of what is analysis and a good background in the mathematical operations behind the theory, I still can't decide wheter to stay with Bartle, right now I have Baby Rudin, Zorich, Pugh, Tao and Kolmogorov but I can't decide of what serves the best for me, what advice could you give on this? Thanks in advace!

[@JoelRosenfeld]

Personally, my favorite text is Rudin’s Principles of Mathematical Analysis. But you can’t go wrong with Bartle. Really, what is important is that you pick a book and stick with it. If you have too many texts on the same topic that you are reading through, then every time you get stuck, you’ll be tempted to go looking through your other texts. If you stick with one, then you’ll be forced to work through those difficulties yourself, which will help you in the long run. For a good measure theory book, I really love Folland’s Real Analysis text. He also has a PDE book that you might be interested in, as a physicist.

[@ashishKjr]

At 3:45 you say that Compactness gives us Functional analysis and Measures in a certain way. Is this a reference to Riesz Representation Theorem for measures (the dual of continuous functions on a locally compact Hausdorff space X that vanish at infinity is the Banach space of all regular complex measures on X)?

[@JoelRosenfeld]

Yep, exactly this. You can manufacture measure theory from functional analysis via the Riesz theorem.

[@NR-ft6cj]

Wake up babe new Math Thing

[@JoelRosenfeld]

Haha! Cheers!

[@fanalysis6734]

I'm not really following the inspiration via finite sets. The image of an arbitrary function on a finite set is finite. The maximum of an arbitrary function on a finite set is finite

[@JoelRosenfeld]

I probably could have explained that better. Essentially, when analysts work with compact sets, we have in the back of our minds that this is a way to extract some “finiteness.” Finite sets are also compact, but we want to work with more general sets, and the right way to expand the collection of finite sets is through compact sets. Where compact sets still let continuous functions take a largest value. It’s also something we see continued later. By leveraging ideas of compact sets, we can see that Compact Operators are the right extension of Finite Matrices.