Tag Archives: analysis

Assume is locally Lipschitz on , that is, for any , there exists (depending on ) such that for all . Then, for any compact set , there exists a constant (depending on ) such that for all . That … Continue reading →

It turns out that there are two types of Lp interpolation: One is called “Lyapunov’s inequality” which is addressed in this previous blog post. The other one is called Littlewood’s inequality: If , then for any intermediate . The proof … Continue reading →

The generalized Riemann-Stieltjes integral is a number such that: for every there exists a partition of such that if , , with is a tagged partition of such that is a refinement of , then We will write and .

Let be a subset of with , and let be a collection of cubes covering . Then there exist a positive constant (depending only on ), and a finite number of disjoint cubes in such that (We may take .)

Assume is a continuously differentiable function on and is a function of bounded variation on . Then is also a function of bounded variation on . Proof: where By Mean Value Theorem, for some . Since is continuous, it is … Continue reading →

Consider . Note that , and Thus , and is improper Riemann integrable. However note that which diverges as (harmonic series). Thus is not Lebesgue integrable on .

This key theorem showcases the full power of Lebesgue Integration Theory. Generalized Lebesgue Dominated Convergence Theorem Let and be sequences of measurable functions on satisfying a.e. in , a.e. in , and a.e. in . If and , then . Proof We … Continue reading →

“Differentiating under the Integral” is a useful trick, and here we describe and prove a sufficient condition where we can use the trick. This is the Measure-Theoretic version, which is more general than the usual version stated in calculus books. … Continue reading →

Assume , where . Consider Thus For , . The above set is equal to , so . as . Hence converges to everywhere.

Cauchy Product: The Cauchy product of two infinite series is defined by where .

Mertens’ Theorem Let and be real or complex sequences. If the series converges to and converges to , and at least one of them converges absolutely, then their Cauchy product converges to . An immediate corollary of Mertens’ Theorem is … Continue reading →

Tietze Extension Theorem If is a normal topological space and is a continuous map from a closed subset , then there exists a continuous map with for all in . Moreover, may be chosen such that , i.e., if is … Continue reading →

Let . Then each of the following four assertions is equivalent to the measurability of . (Outer Approximation by Open Sets and Sets) (i) For each , there is an open set containing for which . (ii) There is a … Continue reading →

Excision property of measurable sets (Proof) If is a measurable set of finite outer measure that is contained in , then Proof: By the measurability of , Since , we have the result.

The Divergence Theorem: is a rather formidable looking formula that is not so easy to memorise. One trick is to remember it is to remember the simpler-looking General Stoke’s Theorem. One can use the general Stoke’s Theorem () to equate … Continue reading →

Source: http://www.math.caltech.edu/~dinakar/08-Ma1cAnalytical-Notes-chap.2.pdf The above paragraph describes nicely the intuitive meaning of the idea behind the definition of differentiability in higher dimensions! It is a very neat idea.

Borel measurability A function is said to be Borel measurable provided its domain is a Borel set and for each , the set is a Borel set. Borel set A Borel set is any set in a topological space that … Continue reading →

Check out this video by “patrickJMT”. His videos are excellent. In this case, the reason why we can change the order of integration is Tonelli’s Theorem, since the integrand is non-negative.

BV functions of one variable Total variation The total variation of a real-valued function , defined on an interval , is the quantity where the supremum is taken over the set . BV function . Jordan decomposition of a function … Continue reading →

One way to remember Markov’s Inequality (also called Chebyshev’s Inequality) is to remember this application: No more than 1/5 of the population can have more than 5 times the average income. For instance, if the average income of a certain … Continue reading →

Bartle’s proof of the Riesz-Fischer theorem ( spaces are complete) is quite nice. Royden uses a concept called “rapidly Cauchy”, which may complicated things unnecessarily. Proof from Bartle’s Book Elements of Integration:

Fatou’s Lemma Let be a sequence of nonnegative measurable functions, then A brilliant graphical way to remember Fatou’s Lemma (taken from the site http://math.stackexchange.com/questions/242920/what-are-some-tricks-to-remember-fatous-lemma). The first two are ∫f1 and ∫f2 respectively, but even the smaller of these is larger than the area … Continue reading →