MT322 Analysis
Spring 2014

Dan Chambers   365 Carney

chambers@bc.edu    617-552-3769 (email is more reliable)

Office Hours: Mondays 12-1, Wednesday 2-3, Fridays 11-12, or by appointment.

This is a followup to MT321, Analysis I. Building on that material, we will cover integral calculus (Riemann and Riemann-Stieltjes integrals), sequences of functions (including uniform convergence of these), the Stone-Weierstrass theorem, power series, exponential and logarithmic functions. Time permitting, we will look at further topics, such as Fourier series or Lebesgue integration.

The text we'll continue to use is Principles of Mathematical Analysis, third edition, by Walter Rudin.

Here is a link to the syllabus.

This is a work in progress; check back for updates.

 Date topic/section HW assigned HW due/solutions January 13 definition of Riemann and R-S integrals a non-integrable function 15 examples of R-S integrals upper and lower sum and refinement results central tool of R-S integrability 17 continuity implies R-S integrability 20 MLK Day 22 monotonicity and integrability piecewise cts implies integrability compositions HW#1 24 an example properties of R-S integrals 27 products of integrable functions R-S integral wrt a step function 29 R-S integral wrt a differentiable function start of FTC, I HW#1 31 FTC, integration by parts, Schwarz inequality HW#2 February 3 sequences and series of functions, introduction 5 snow day- BC canceled 7 uniform convergence- sequences 10 uniform convergence- series; Abel's Theorem HW#3 HW#2 12 uniform convergence and integrability alternating harmonic series and \pi/4 14 uniform convergence and continuity sup norm 17 R(X) as a complete metric space uniform convergence and differentiability HW#3 19 exam 1  solns 21 more on uniform convergence and differentiability 24 a continuous everywhere, differentiable nowhere function 26 uniform boundedness and equicontinuity of a sequence of functions 28 convergence of a subsequence of functions HW#4 March 3 spring break 5 spring break 7 spring break 10 approximating a function; Bernstein polynomials 12 proof of Weierstrass's approximation theorem 14 conclusion of proof; power series introduction HW#4 17 power series: continuity, differentiability, integrability results a series representation for pi 19 more power series manipulations intro to Taylor/MacLaurin series HW#5 21 functions and their T/M series 24 convergence of the M. series of f to f 26 E(x) vs e^x, L(x) vs ln(x) HW#6 HW#5 28 why does L(E(x))=x? a series answer start of orthonormal sets, inner product spaces 31 inner product spaces, C-S inequality April 2 start of Fourier series HW#6 4 exam 2  solns 7 Fourier series, examples, convergence 9 F. series wrap up; functions of several variables: linear transformations 11 linear transformations and differentiation 14 differentiation continued 16 inverse function theorem 18 Easter break 21 Patriots' Day HW#7 23 Lebesgue theory, sigma-algebras, measurable sets 25 measurable sets, measurable functions 28 measurable functions HW#7 30 Lebesgue integrals May 7 final exam 12:30 pm