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Sobolev space - Wikipedia
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself and its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space. Intuitively, a Sobolev space is a space of functions with sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function.
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february 2017 by nhaliday
A VERY BRIEF REVIEW OF MEASURE THEORY
A brief philosophical discussion:
Measure theory, as much as any branch of mathematics, is an area where it is important to be acquainted with the basic notions and statements, but not desperately important to be acquainted with the detailed proofs, which are often rather unilluminating. One should always have in a mind a place where one could go and look if one ever did need to understand a proof: for me, that place is Rudin’s Real and Complex Analysis (Rudin’s “red book”).
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february 2017 by nhaliday
Cauchy-Schwarz inequality and Hölder's inequality - Mathematics Stack Exchange
- Cauchy-Schwarz (special case of Holder's inequality where p=q=1/2) implies Holder's inequality
- pith: define potential F(t) = int f^{pt} g^{q(1-t)}, show log F is midpoint-convex hence convex, then apply convexity between F(0) and F(1) for F(1/p) = ||fg||_1
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january 2017 by nhaliday
Dvoretzky's theorem - Wikipedia
In mathematics, Dvoretzky's theorem is an important structural theorem about normed vector spaces proved by Aryeh Dvoretzky in the early 1960s, answering a question of Alexander Grothendieck. In essence, it says that every sufficiently high-dimensional normed vector space will have low-dimensional subspaces that are approximately Euclidean. Equivalently, every high-dimensional bounded symmetric convex set has low-dimensional sections that are approximately ellipsoids.

http://mathoverflow.net/questions/143527/intuitive-explanation-of-dvoretzkys-theorem
http://mathoverflow.net/questions/46278/unexpected-applications-of-dvoretzkys-theorem
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january 2017 by nhaliday
cv.complex variables - Absolute value inequality for complex numbers - MathOverflow
In general, once you've proven an inequality like this in R it holds automatically in any Euclidean space (including C) by averaging over projections. ("Inequality like this" = inequality where every term is the length of some linear combination of variable vectors in the space; here the vectors are a, b, c).

I learned this trick at MOP 30+ years ago, and don't know or remember who discovered it.
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january 2017 by nhaliday
Cantor function - Wikipedia
- uniformly continuous but not absolutely continuous
- derivative zero almost everywhere but not constant
- see also: http://mathoverflow.net/questions/31603/why-do-probabilists-take-random-variables-to-be-borel-and-not-lebesgue-measura/31609#31609 (the exercise mentioned uses c(x)+x for c the Cantor function)
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january 2017 by nhaliday

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