real analysis Consider these two mappings that is functions

real analysis

Consider these two mappings (that is, functions) from the set X = C[-1, 1] into the real numbers R : T(f) = integral_-1^1 f(x) dx and D(f) = f(0). For each of these two functions, and for each of the two metric spaces (X, d_infinity) and (X, d_1), (that\'s 4 questions overall), ask this: is the function in question a continuous function from the metric space in question to the metric space R? Prove your answers.

Solution

Real analysis is a large field of mathematics that is entirely based on the properties of the real numbers and the ideas of sets, functions, and limits. The base of this theorem is behind behind calculus, differential equations, and probability specially. Real analysis helps us for an appreciation of the many interconnections with other mathematical areas.

In other words real Analysis is an enormous field having its applications to many areas of mathematics. It has applications to setting where one integrates functions, ranging from harmonic analysis on Euclidean space to partial differential equations on manifolds, from representation theory to number theory, from probability theory to integral geometry, from ergodic theory to quantum mechanics.

Thus in short, real analysis is anyhow a theoretical field that is closely related to mathematical concepts used in most branches of economics such as calculus and probability theory. It is a theory of functions of a real variable.