AMATH 567: Applied Analysis

Course Description

Textbook

Notes on numerical solution of ODE's: .pdf

Syllabus

(1) Norms and Metric Spaces:

Metrics, norms, convergence of sequences and series, Cauchy sequences, completeness, upper and lower bounds, continuity, open and closed sets, compactness, maxima and minima.

(2) Continuous Functions:

Spaces of continuous functions, approximation by polynomials -- Weierstrass approximation theorem, equicontinuous functions -- Arzela-Ascoli theorem.

(3) Initial Value Problem for Ordinary Differential Equations:

Existence and uniqueness of solutions, contraction mapping fixed point theorem, Gronwall's inequality.

(4) Banach Spaces:

Bounded linear operators, kernel and range of a linear map, finite dimensional Banach spaces, matrix of a linear transformation.

(5) Bounded Linear Operators on a Hilbert Space:

Orthogonal projection, Fourier series, Riesz representation theorem, spectrum of a bounded linear operator.

(6) Introduction to Distributions:

Delta distribution, linear differential equations and Green's functions.

Learning objectives and instructor expectations

Schedule and Homework

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Class Summaries

class summaries

Grading

Tutorials