Math 131a: Real Analysis
نویسنده
چکیده
Real analysis is a rigorous study of the (hopefully familiar) objects from calculus: the real numbers, functions, limits, continuity, derivatives, sequences, series, and integrals. While you are likely quite familiar with computing with these objects, this course will focus on developing the theoretical foundation for the definitions, theorems, formulas, and algorithms that you are used to using. We will start by building up the real numbers from scratch, i.e., from just a few basic axioms, then we will focus our attention on proving many of the things you already believe about functions, sequences, and series. Along the way we will encounter several “pathological” objects which will hopefully convince you that our careful approch is necessary and worthwhile. To get an idea of how subtle some questions in analysis can be, ask yourself: what is a real number? We will answer this question in due time, but for now let’s focus on one specific real number that got a lot of attention from the ancient Greeks: √ 2. Prior to the discovery that √ 2 is an irrational number, it was assumed that: given any two line segments AB and CD, there is a rational number r so that the length of CD equals r times the length of AB. However, the length of the diagonal of a square of side length 1 (using the Pythagorean Theorem) equals √ 2, so by the previous assumption, we must have that √ 2 is a rational number. The proof of the following theorem is one of the most classic proofs in mathematics.
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تاریخ انتشار 2017