Actual analysis

Fourier Analysis: Approximation of a discontinuous function through a punctually convergent series of senoidal functions.

The real analysis or theory of functions of real variables is the branch of mathematical analysis that deals with the set of real numbers and functions of numbers real. In particular, he studies the analytical properties of functions and sequences of real numbers; its limit, continuity and the calculation of real numbers.

Construction of real numbers

Theorems of real analysis are based on the properties of the real numbers system, which must be established. The system of real numbers consists of an uncountable set (${displaystyle mathbb {R} }$), together with two binary operations called + and ⋅and an order called .. The operations make the real numbers a field, and, along with the order, an orderly field. The real number system is the only field ordered CompleteIn the sense that any other ordained field is isomorphous to it. Intuitively, completeness means that there are no "bones" in real numbers. This property distinguishes real numbers from other ordered fields (e.g., rational numbers) ${displaystyle mathbb {Q} }$) and is fundamental to the demonstration of several key properties of the functions of the real numbers. The completeness of the real is often expressed conveniently as the property of the minimum upper limit.

Scope

Real analysis is an area of mathematical analysis that studies the concepts of sequence, limit, continuity, differentiation, and integration. Given its nature, real analysis is limited to real numbers as working tools.

Important results include among others the Bolzano-Weierstrass theorem, the Heine-Borel theorem, the mean value theorem, and the fundamental theorem of calculus.

Basics

"Advanced calculus" texts usually begin with an introduction to mathematical proofs and set theory. After this, the real numbers are defined axiomatically, or they are constructed with Cauchy sequences or as Dedekind cuts of rational numbers. Next, they investigate the properties of real numbers, one of the most important being the triangular inequality.

Sequences and series

After defining the real numbers, we investigate the sequences of real numbers and their convergence, a central concept in analysis, through the limits of sequences or accumulation points of sets. Subsequently, series are studied, such as alternate series and power series.

In order to begin to develop elementary topological concepts, various types of subsets of the real numbers are studied: open sets, closed sets, compact spaces, connected sets, etc., where the Bolzano-Weierstrass theorem and the theorem of Heine-Borel.

Continuous functions

Now the functions of real variables are studied, and the concept of continuous function is defined from the epsilon-delta definition of the limit of a function. Among the properties of a continuous function defined on an interval, the theorems known as Bolzano's theorem, the intermediate value theorem and the Weierstrass theorem stand out.

Derivation or differentiation

Now you can define the derivative of a function as a limit, and you can rigorously prove important theorems about differentiation like Rolle's theorem or the mean value theorem. The Taylor series are constructed and the Maclaurin series of the exponential functions and the trigonometric functions are calculated.

It is important to note that the functions of various variables as well as their derivatives, which are partial derivatives, are also studied. It is very important to study the inverse function theorem and the implicit function theorem, as well as the Morse functions.

Integration

Definite integration, which can be defined as "the area under the graph" of a function, naturally follows the differentiation, of which indefinite integration is the inverse operation. You start with the Riemann integral, which consists of dividing the interval into subintervals (with a partition), extending the subintervals up until you reach, or the minimum of the function on the subinterval (in which case it is called the lower sum), or to the maximum in the subinterval (in which case it is called the upper sum). There is also another type of integral, which can integrate more functions, called the Lebesgue integral, which uses the measure and the concept of "almost everywhere". The same is shown later.

With the integration theory several theorems can be demonstrated, in the case of Riemann or Lebesgue integration, such as Fubini's theorem, but more importantly the fundamental theorem of calculus.

Back to basics in more general settings

Having done all this, it is useful to return to the concepts of continuity and convergence, and study them in a more abstract context, in preparation for studying spaces of functions, which is done in functional analysis or more specialized analysis such as complex.

Generalizations and Related Areas of Mathematics

Several ideas from real analysis can be generalized from the real line to larger or more abstract contexts. These generalizations link actual analysis to other disciplines and subdisciplines. For example, the generalization of ideas such as continuous functions and compactness from real analysis to metric spaces and topological spaces connects real analysis to the field of general topology, while the generalization of finite-dimensional Euclidean spaces to infinite-dimensional analogues led to the concepts of Banachs spaces and Hilberts spaces and, more generally, functional analysis. Georg Cantor's research on the sets and sequences of real numbers, the mappings between them, and the foundational issues of real analysis gave rise to naive set theory. The study of convergence issues for sequences of functions eventually gave rise to Fourier analysis as a subdiscipline of mathematical analysis. Investigation of the consequences of generalizing the differentiability of functions of a real variable to functions of a complex variable gave rise to the concept of a holomorphic function and to the beginning of complex analysis as another distinct subdiscipline of analysis. On the other hand, the generalization of integration from the Riemannian sense to that of Lebesgue led to the formulation of the concept of abstract measure space, a fundamental concept in measure theory. Finally, the generalization of integration from the real line to curves and surfaces in higher-dimensional spaces gave rise to the study of vector calculus, whose further generalization and formalization played an important role in the evolution of the concepts of differential and smooth multiple forms. (differentiable) in differential geometry and other closely related areas of geometry and topology.