Instructor: Laszlo Erdös
Teaching Assistant: Johannes Alt
How should one look at a real function? What meaningful questions one may ask about them? Why do they play a central role?
The course presents some essentials of modern mathematical analysis to natural scientists. Instead of the traditional presentation of this vast area to mathematicians that focuses on conceptual development, we will have a “hands-on” approach, selecting a few key features that we found useful in mathematical research motivated by applications. A central theme is inequalities, i.e. the idea that exact computations are often not feasible, but estimating a quantity still provides useful information. We will rely on the book of Lieb and Loss, but other topics will also be covered, partly depending on the interests of the students
There will be a written exam at the end of the course.
The final grade is determined based upon the homework exercises and the final exam.
|October 10, 2016:||Heuristic proof of Euler’s formula for the sum of reciprocals of the squares following Euler. Intuitive factorization of the Taylor series of the sine function as an “infinite” polynomial. Convergent and divergent series. Convergent infinite products. Convergence of the sum of 1/n^2, divergence of the sum of 1/n.|
|October 12, 2016:||We gave a rigorous proof of Euler’s formula using a bit trigonometry and integration by parts. We used Riemann-Lebesgue lemma (in a baby form) L’Hospital rule and Taylor expansion in “real” situations.|
|October 17, 2016:||The main concepts of analysis are approximation and closeness (in constrast to algebra that focuses on identities and equalities). To measure closeness, we introduce certain structures on the set (space) of objects we study. Definition of metric and normed spaces. Examples. Continuous functions on [0,1] with its two natural norms (integral and maximum norms). Concept of completeness and completion of an incomplete space demonstrated on the example of root 2.|
|October 19, 2016:||C[0,1], i.e. the continuous functions on [0,1] is a complete metric (even normed) space with the maximum norm. We proved that every Cauchy sequence converges and, most importantly, the limit is continuous. This was done by introducing the “epsilon/3” argument, stressing that interchanging limits is allowed only if one of the limits is uniform. Counterexample was presented for the general (non-uniform) case. The same space C[0,1] with the integral norm is not complete, we have seen counterexamples: Dirichlet function and its natural approximants with finitely many jumps. Definition of the Riemann integral; the Dirichlet function is not Riemann integrable and there is no “cheap” way to extend integration to it. There are (at least) two reasons one wants to have a better concept of integration than Riemann: (i) completeness and (ii) interchangeability of the limit and integral. Trick: when determining the approximations to the “area under the graph of the function”, subdivide the y-axis instead of the usual subdivision of the x-axis. This has the advantage that an appropriate (dyadic) sequence of approximants converge. The main difficulty is that is tacitly assumed that we know the “measure” of every subset of the real line (as they may appear as preimagines). Bad news: there is no such “measure”, that would also satisfy the natural property of additivity and invariance under translation — example. Way out: do not try to “measure” every subset! Then the theory becomes more subtle, but there is no cheaper way.|
|October 24, 2016:||I repeated the non-measurability of the set originating from taking equivalence relation “difference is rational” on [0,1]. Some discussion of Zermelo-Frenkel axioms and the axiom of choice. Some philosophy about its relevance and consequences. Banach Tarski paradox. All these justify that we need to cleverly select a family of “good” sets. Definition of sigma algebra. Borel sigma algebra on R^n. The building blocks can be balls, rectangles, open/closed sets.|
|October 31, 2016:||Sigma algebra, Borel sigma algebra and its existence (discussion on constructive versus nonconstructive arguments) One can use either balls, rectangles or open sets in the definition (all are the same). Elements of the Borel sigma algebra are called measurable sets. Example: the set of rational numbers is measurable. Definition and basic properties of Lebesgue measure (invariance under Euclidean motions). Measurable functions. Two definitions of Lebesgue integral. Properties: additivity, monotone convergence, consistency with Riemann integral. Dominated convergence theorem|
|November 2, 2016:||Definition of the Lebesgue integral for functions that have both positive and negative parts. Discussions of infinity minus infinity and the impossibility to define improper integrals in a stable way. Completion of C[0,1] under the integral norm is the Lebesgue integrable functions. Minimality of the completion. Riesz-Fischer theorem (statement only). Zero measure sets, Lebesgue sigma algebra.|
|November 7, 2016:||Zero measure sets. Equivalence class of functions, concept of “with the exception of a zero measure set”. Proper definition of L^p spaces (including p= infinity). Discussion of examples showing that L^p is sensitive to strong singularities for large p and slow decay tail at infinity for small p. Riesz Fischer theorem (proof next time). Completion of C[0,1] and Riemann integrable functions with the integral norm. L^infinity is not the completion of C[0,1] with the maximum norm, since that is already complete, but this is a special case, for all finite p, L^p is the “good” space.|
|November 9, 2016:||Proof of Riesz Fischer theorem. Relation between convergence in L^p and pointwise. Convexity and strict convexity. Jensen’s inequality.|
|November 14, 2016:||Probability space as a special measure space. Holder inequality with proof. Meaning of Holder: decoupling. Minkowski inequality with proof. Young’s inequality (without proof). Definition and symmetry of convolution. Its meaning in case when one function is an approximate delta function: smoothing out the other function. Application: any L^p function (with p< infinity) is approximable by smooth, compactly supported functions in L^p sense. This completes the proof that the completion of continuous (or even smooth) functions in L^p sense is the L^p space.|
|November 16, 2016:||Hamiltonian formalism of classical mechanics in nutshell. Instability of the H-atom in classical mechanics. Basic setup of quantum mechanics, wave function, observables, position, momentum, Hamilton operator, Schrodinger equation. Stability of H-atom in QM. Main lemma: kinetic energy controls the potential energy in case of a Coulomb interaction.|
|Homework 1||October 24, 2016|
|Homework 2||November 7, 2016|
|Homework 3||November 21, 2016|
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