Introduction to Analysis Fall 2017
Math 104 (Lecture 006) at UC Berkeley
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About the Course
Is there a rational number between any two real numbers? ... and what are the real numbers?
Are there more rational or more irrational numbers? ... and what does “more” mean?
Is every continuous function differentiable? ... or at least in most points? ... or maybe sometimes nowhere at all?
What does convergence mean? ... and what does it take to check if a sequence (of numbers, functions, ...) converges? ... and what if a sequence does not converge?
Is a function with (basically everywhere) vanishing derivative constant?
At a local maximum, a differentiable function has vanishing derivative. But is a differentiable function non-decreasing close to (but on the left of) a local max?
These basic questions are usually not explained in calculus classes. This course is intended to be a preparation for real analysis, the study of functions of one real variable. We will prove (and sometimes disprove) "facts" which you might have taken for granted up to now and will discuss some of the questions above. Although many concepts discussed in the class (convergence of a sequence, continuity of a function,...) may be familiar from calculus classes, the rigorous, proof-based approach of this course is challenging and shows somewhat different facets of mathematics. Proofs in analysis are different from those you might know from other proof-based classes. One main goal of the course is to teach you how to read proofs, how to come up with the idea of a proof, and finally, to write it down so that other people can follow your idea. We will start by introducing real and complex numbers, and Euclidean spaces. Then metric spaces, a generalization of these concepts, will be introduced. An important point of the course is a proper understanding of convergence of sequences (and series), which will build the basis for understanding functions. In particular, we will study continuity and differentiability of functions. The Riemann integral will be discussed and sequences of functions will be analyzed, and we will be able to prove the fundamental Arzelà-Ascoli theorem and the theorems of Stone and Weierstra?.
Lectures
Mo We Fr 12-1pm in 310 Hearst Mining.
Office Hours
You can find me in 895 Evans Hall