Functional analysis is a branch of mathematics that studies the properties of functions and their relationship to linear operators and other transformations. It has applications in the fields of engineering, physics, economics, and other sciences.
Definition
Functional analysis is a branch of mathematics that examines the properties of a function and its relationship to linear operators and other transformations. It utilizes the concepts of mathematical analysis, such as differential and integral calculus, to study the behavior of functions in various domains. It is also used in the study of partial differential equations and other mathematical models.
History
Functional analysis originated with the work of mathematicians such as Leibniz, Euler, and Cauchy in the 17th and 18th centuries. It was further developed in the 19th century by mathematicians such as Weierstrass, Riemann, and Hilbert. Later, it was extended by mathematicians such as Lévy, Sobolev, and Banach in the 20th century. Functional analysis has been used to solve a variety of problems in mathematics, physics, engineering, and economics.
Characteristics
Functional analysis is characterized by the use of linear operators and their properties, such as continuity, differentiability, and compactness. It is also characterized by the use of topological spaces and their properties, such as connectedness, compactness, and continuity. Furthermore, it is characterized by the use of measure theory and its properties, such as integration, differentiation, and convergence.
References
Rudin, W. (1987). Real and complex analysis (3rd ed.). New York: McGraw-Hill.
Lax, P. (2002). Functional analysis. New York: Wiley.
Kreyszig, E. (2011). Introductory functional analysis with applications (2nd ed.). Hoboken, NJ: Wiley.
Weber, K. (2011). Functional Analysis: An Introduction. New York: Springer.
Strang, G. (2009). Introduction to linear algebra (4th ed.). Wellesley, MA: Wellesley-Cambridge Press.