Real Analysis

Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. It can be seen as a rigorous version of calculus and studies concepts such as sequences and their limits, continuity, differentiation, integration and sequences of functions. It is a sophisticated theory of the ‘numerical function’ idea, and contains modern theories of generalized functions.

The presentation of real analysis in advanced texts usually starts with simple proofs in elementary set theory, a clean definition of the concept of function, and an introduction to the natural numbers and the important proof technique of mathematical induction.

Then the real numbers are either introduced axiomatically, or they are constructed from Cauchy sequences or Dedekind cuts of rational numbers. Initial consequences are derived, including the concept of the least upper bound and the properties of the absolute value such as the triangle inequality and Bernoulli’s inequality.

The concept of convergence, central to analysis, is introduced via limits of sequences. Several laws governing the limiting process can be derived, and several limits can be computed. Infinite series, which are special sequences, are also studied at this point. Power series serve to cleanly define several central functions, such as the exponential function and the trigonometric functions. Various important types of subsets of the real numbers, such as open sets, closed sets, compact sets and their properties are introduced next, such as the Bolzano-Weierstrass and Heine-Borel theorems.

The concept of continuity may now be defined via limits. One can show that the sum, product, composition and quotient of continuous functions is continuous, excluding at points where the denominator function has value zero, and the important intermediate value theorem is proven. The notion of derivative may be introduced as a particular limiting process, and the familiar differentiation rules from calculus can be proven rigorously. A central theorem here is the mean value theorem.

Then one can do integration (Riemann and Lebesgue) and prove the fundamental theorem of calculus, typically using the mean value theorem.

At this point, it is useful to study the notions of continuity and convergence in a more abstract setting, in order to later consider spaces of functions. This is done in point set topology and using metric spaces. Concepts such as compactness, completeness, connectedness, uniform continuity, separability, Lipschitz maps, contractive maps are defined and investigated.

One can take limits of functions and attempt to change the orders of integrals, derivatives and limits. The notion of uniform convergence is important in this context. Here, it is useful to have a rudimentary knowledge of normed vector spaces and inner product spaces. Taylor series can also be introduced here.

Some introductory real analysis courses (and their textbooks) are significantly different in scope: for example, Lebesgue integration is not always covered, while Riemann-Stieltjes integration sometimes is.

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