REAL ANALYSIS I
Prerequisite and degree relevance: Either consent of Mathematics Advisor, or two of M341, 328K, 325K (Philosophy 313K may be substituted for M325K), with a grade of at least C-.
Students who receive a grade of C in M325K or M328K are advised to take M361K before attempting M365C.
Course description: This course is an introduction to Analysis. Analysis together with Algebra and Topology form the central core of modern mathematics. Beginning with the notion of limit from calculus and continuing with ideas about convergence and the concept of function that arose with the description of heat flow using Fourier series, analysis is primarily concerned with infinite processes, the study of spaces and their geometry where these processes act and the application of differential and integral to problems that arise in geometry, pde, physics and probability. This should be a course in analysis rather than point-set topology; the latter is covered in M376K.
Text: An appropriate text is Rudin "Principles of Mathematical Analysis" and the course should roughly cover its first seven chapters. The main difference between M361K and M365C lies in the more abstract metric space point of view in the latter. A strong student should be able to handle M365C without first taking M361K.
- The real number system and Euclidean spaces: the axiomatic description of the real number system as the unique complete ordered field; the complex numbers; Euclidean space IR .
- Metric spaces: elementary metric space topology, with special emphasis on Euclidcan spaces; sequences in metric spaces - limits, accumulation points, subsequences, etc.; Cauchy sequences and completeness; compacmess in metric spaces; compact sets in R; connectedness in metric spaces; countable and uncountable sets.
- Continuity: limits and continuity of mappings between metric spaces, with particular attention to real-valued functions def'med on subsets of IR; preservation of compactness and connectedness under continuous mapping; uniform continuity.
- Differentiation on the line: the definition and geometric significance of the derivative of a real-valued function of a real variable; the Mean Value Theorem and its consequences; Taylor's Theorem; L'Hospital's rules.
- Riemann integration on the line: the definition and elementary properties of the Riemann integral; existence theorems for Riemann integrals; the Fundamental Theorems of Calculus.
- Sequences and series of functions: uniform convergence, uniform convergence and continuity, uniform convergence and integration, uniform convergence and differentiation.