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Tag Archives: analysis
Locally Lipschitz implies Lipschitz on Compact Set Proof
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
Lp Interpolation
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
Generalized Riemann-Stieltjes Integral
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 .
Sufficient condition for “Weak Convergence”
This is a sufficient condition for something that resembles “Weak convergence”: for all Suppose that a.e.\ and that , . If , we have for all , . Note that the result is false if . Proof: (Case: , where … Continue reading
Square root x is not Lipschitz on [0,1]
is not Lipschitz on : Suppose there exists such that for all , , By Mean Value Theorem, this means that for some between and . However, is unbounded on , a contradiction. Note however, that is absolutely continuous on … Continue reading
Young’s Convolution Theorem
Let and , and let . If and , then and Amazing Theorem! If , then .
Relationship between L^p convergence and a.e. convergence
It turns out that convergence in Lp implies that the norms converge. Conversely, a.e. convergence and the fact that norms converge implies Lp convergence. Amazing! Relationship between convergence and a.e. convergence: Let , . If , then . Conversely, if … Continue reading
Simple Vitali Lemma
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 .)
Wheeden Zygmund Measure and Integration Solutions
Here are some solutions to exercises in the book: Measure and Integral, An Introduction to Real Analysis by Richard L. Wheeden and Antoni Zygmund. Chapter 1,2: analysis1 Chapter 3: analysis2 Chapter 4, 5: analysis3 Chapter 5,6: analysis4 Chapter 6,7: analysis5 Chapter 8: analysis6 Chapter 9: analysis7 Measure … Continue reading
Absolute Continuity of Lebesgue Integral
The following is a wonderful property of the Lebesgue Integral, also known as absolute continuity of Lebesgue Integral. Basically, it means that whenever the domain of integration has small enough measure, then the integral will be arbitrarily small. Suppose is … Continue reading
Inequalities for pth powers, where 0<p<infinity
There are some useful inequalities for , where p is a number ranging from 0 to infinity. These are the top 3 useful inequalities (note some of them only work for p less than 1, or p greater than 1). … Continue reading
Composition of Continuously Differentiable Function and Function of Bounded Variation
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
Fatou’s Lemma for Convergence in Measure
Suppose in measure on a measurable set such that for all , then . The proof is short but slightly tricky: Suppose to the contrary . Let be a subsequence such that (using the fact that for any sequence there … Continue reading
Summation by parts / Abel’s Lemma
This is an amazing identity by Abel. Let and be two sequences. Then,
sin(1/x)/x is improper Riemann Integrable but not Lebesgue Integrable on (0,1]
Consider . Note that , and Thus , and is improper Riemann integrable. However note that which diverges as (harmonic series). Thus is not Lebesgue integrable on .
Lebesgue’s Dominated Convergence Theorem for Convergence in Measure
Lebesgue’s Dominated Convergence Theorem for Convergence in Measure If satisfies on and , then and . Proof Let be any subsequence of . Then on . Thus there is a subsequence a.e.\ in . Clearly . By the usual Lebesgue’s … Continue reading
Generalized Lebesgue Dominated Convergence Theorem Proof
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
Leibniz Integral Rule (Differentiating under Integral) + Proof
“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
Laurent Series with WolframAlpha
WolframAlpha can compute (simple) Laurent series: https://www.wolframalpha.com/input/?i=series+sin(z%5E-1) Series[Sin[z^(-1)], {z, 0, 5}] 1/z-1/(6 z^3)+1/(120 z^5)+O((1/z)^6) (Laurent series) (converges everywhere away from origin) Unfortunately, more “complex” (pun intended) Laurent series are not possible for WolframAlpha.
Increasing sequence of simple functions to a bounded measurable function f
Assume , where . Consider Thus For , . The above set is equal to , so . as . Hence converges to everywhere.
Laurent Series (Example)
The Laurent series is something like the Taylor series, but with terms with negative exponents, e.g. . The below Laurent Series formula may not be the most practical way to compute the coefficients, usually we will use known formulas, as … Continue reading
Implicit Function Theorem
The implicit function theorem is a strong theorem that allows us to express a variable as a function of another variable. For instance, if , can we make the subject, i.e. write as a function of ? The implicit function … Continue reading
lim sup & lim inf of Sets
The concept of lim sup and lim inf can be applied to sets too. Here is a nice characterisation of lim sup and lim inf of sets: For a sequence of sets , consists of those points that belong to … Continue reading
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus is one of the most amazing and important theorems in analysis. It is a non-trivial result that links the concept of area and gradient, two seemingly unrelated concepts. Fundamental Theorem of Calculus The first part … Continue reading
Gradient Theorem (Proof)
This amazing theorem is also called the Fundamental Theorem of Calculus for Line Integrals. It is quite a powerful theorem that sometimes allows fast computations of line integrals. Gradient Theorem (Fundamental Theorem of Calculus for Line Integrals) Let be a … Continue reading
Liouville’s Theorem
Liouville’s Theorem Every bounded entire function must be constant. That is, every holomorphic function for which there exists such that for all is constant.
Multivariable Version of Taylor’s Theorem
Multivariable calculus is an interesting topic that is often neglected in the curriculum. Furthermore it is hard to learn since the existing textbooks are either too basic/computational (e.g. Multivariable Calculus, 7th Edition by Stewart) or too advanced. Many analysis books skip multivariable calculus … Continue reading
Cauchy Product Definition
Cauchy Product: The Cauchy product of two infinite series is defined by where .
Pasting Lemma (Elaboration of Wikipedia’s proof)
The proof of the Pasting Lemma at Wikipedia is correct, but a bit unclear. In particular, it does not clearly show how the hypothesis that X, Y are both closed is being used. It actually has something to do with subspace … Continue reading
One-Sided Limit that Does Not Exist
Offhand, it is hard to think of a function that does not have even a one-sided limit. This video shows one!
Algebra and Analysis Theorems
The following are two lists of useful algebra and analysis theorems that are covered during university. Algebra Theorems Mathtuition88 Analysis Theorems Mathtuition88
Mertens’ Theorem
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 and Pasting Lemma
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
Lusin’s Theorem and Egorov’s Theorem
Lusin’s Theorem and Egorov’s Theorem are the second and third of Littlewood’s famous Three Principles. There are many variations and generalisations, the most basic of which I think are found in Royden’s book. Lusin’s Theorem: Informally, “every measurable function is … Continue reading
A holomorphic and injective function has nonzero derivative
This post proves that if is a function that is holomorphic (analytic) and injective, then for all in . The condition of having nonzero derivative is equivalent to the condition of conformal (preserves angles). Hence, this result can be stated … Continue reading
Underrated Complex Analysis Theorem: Schwarz Lemma
The Schwarz Lemma is a relatively basic lemma in Complex Analysis, that can be said to be of greater importance that it seems. There is a whole article written on it. The conditions and results of Schwarz Lemma are rather … Continue reading
Groups of order pq
In this post, we will classify groups of order pq, where p and q are primes with p<q. It turns out there are only two isomorphism classes of such groups, one being a cyclic group the other being a semidirect … Continue reading
Rouche’s Theorem
Rouche’s Theorem If the complex-valued functions and are holomorphic inside and on some closed contour , with on , then and have the same number of zeroes inside , where each zero is counted as many times as its multiplicity. … Continue reading
The most Striking Theorem in Real Analysis
Lebesgue’s Theorem (see below) has been called one of the most striking theorems in real analysis. Indeed it is a very surprising result. Lebesgue’s Theorem (Monotone functions) If the function is monotone on the open interval , then it is … Continue reading
Inner and Outer Approximation of Lebesgue Measurable Sets
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 in Measure Theory
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.
How to remember the Divergence Theorem
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
Why Differentiability in Higher Dimensions is defined as it is?
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
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
How to Change Order of Integration
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.
Lebesgue’s Dominated Convergence Theorem (Continuous Version)
This is a basic but very useful corollary of the usual Lebesgue’s Dominated Convergence Theorem. From what I see, it is basically the Sequential Criterion plus the usual Dominated Convergence Theorem. From the book: Basic Partial Differential Equations
BV (Bounded Variation) functions
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
Markov’s Inequality: No more than 1/5 of the population can have more than 5 times the average income
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
Riesz-Fischer Theorem Proof
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
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
