Instructor:

Robert J. Vanderbei, ACE-42, 8-0876

Lectures:

1:30-2:20 MWF, E-223 (tentative)

Office Hours:

3:00-6:00 W, ACE-42

1 Preface

The goal of this course is to cover the basics of analysis in a form suitable for engineers. The aim is to prepare students for advanced courses in differential equations, stochastics, optimization and related topics.

The students are assumed to have some maturity: they should be familiar with linear algebra and ordinary calculus. The exercises are provided with three aims:

• to push the student to learn by doing;
• to help him/her learn to present results in a clear, lucid style;
• to provide examples and facts that are often useful by themselves.

The course starts with the primitive notions and a review of the properties of real numbers. Then, ideas of closeness, distance, convergence, and continuity are developed by way of metric spaces. These notions are then used to develop the derivative and its properties. Next, we cover the theory of integration, and its basic tools, in abstract measure spaces. Finally, fixed point theory, integral equations, Fourier analysis, and convergence of certain algorithms will be covered.

There will be weekly assignments and a final exam. The assignments will count for 75% of the final grade and the final exam will count for the other 25%.

2 Outline

• Sets and Functions
  • Sets, set operations, real line, infimum and supremum
  • Functions, injections and surjections, positivity, etc.
  • Sequences, limits, convergence of sequences, series
  • Cardinality, countability, uncountable sets
• Metric Spaces
  • Metrics, norms (Manhattan Metric, l²-norm, weighted l²-norm, etc.)
  • Open and closed sets, interior, closure, boundary
  • Convergence of sequences, completeness, compactness
  • Continuous functions, Lipschitz continuity, uniform continuity
  • Sequences of functions, pointwise convergence, uniform convergence
  • Spaces of continuous functions, norms, metrics, functionals
• Differentiation
  • Definition, mean value theorem, L’Hopital’s rule
  • Derivatives of higher order and Taylor’s theorem
  • Convexity and subdifferentials
• Applications to Differential and Integral Equations
  • Fixed point theorem, method of successive approximations
  • Systems of linear equations, conditions for existence and uniqueness of solutions (using different metrics)
  • Fredholm and Volterra equations, kernels, resolvants
  • Systems of differential equations, Picard’s method
• Measure and Integration
  • Algebras, measurable spaces and functions
  • Measures, the Lebesgue measure, transformations of measures
  • Integrals, limit theorems
  • Riemann integrals, Stieltjes integrals, line and surface integrals
  • Change of variables, derivatives of measures, differentiation
  • Transition kernels, iterated integrals, measure-kernel-function setup
• Function Spaces
  • Normed vector spaces, linear operators and functionals
  • Hilbert spaces, orthogonality, projections
  • Fourier series
  • Legendre, Hermite, and Laguerre polynomials
  • Riesz representation theorem
  • Adjoint, Hermitian, unitary and normal operators

3 References

The following texts have been put on reserve in the Engineering library:

• W. RUDIN, Principles of Mathematical Analysis
• G. STRANG, Linear Algebra and Its Applications
• A. KOLMOGOROV and S. FOMIN, Introductory Real Analysis
• E. KREYSZIG, Introductory Functional Analysis with Applications
• E. KREYSZIG, Advanced Engineering Mathematics
• G. STRANG, Introduction to Applied Mathematics