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Functional Analysis

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The mathematics is a vast subject which is classified into several branches. The functional analysis is one of most important branches studied in higher mathematics. It deals with the processes of developing as well as analyzing the structure of a function.

It also studies about functional spaces. In pure and theoretical mathematics, it serves as an important tool. In functional analysis, we learn method which explain about processing of complex system.
According to strategies of functional analysis, such processing are defined by the getting a decomposed form of given complex function into small sub-functions which are generally quite simpler than the given function. In this way, it becomes easier to explain complex system.
We shall go ahead and learn about the basics of functional analysis in this page. We shall also know about normed vector spaces and general applications of functional analysis.

Definition

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The functional analysis is a mathematical branch dealing with the properties of classes and mappings of functions. It is concerned with the vector spaces (usually infinite-dimensional functional spaces) and their mappings. Such mappings are known as operators or functionals. Basically, functional analysis studies functional spaces and properties of functional transformations, for example - Fourier transformations.

The functional analysis uses the combination of methods of algebra, topology and classical analysis.

The study of functional analysis generally falls into three basic categories:

1) Introduction of infinite-dimensional spaces

2) The functions that take values in infinite-dimensional spaces.

3) The study of linear functions, most commonly linear operators.

This subject is assumed to be started in the early 19th century and had established finally in the third and fourth decade of 19th century. There was great influence of quantum mechanics on the development of functional analysis.

Normed Vector Spaces

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The normed vector spaces are the basic spaces that are being studied in functional analysis. They are generally defined over real or complex numbers. They are also known as Banach spaces.

The normed vector spaces have a great importance in several fields including mathematical formulation of quantum mechanics. The functional analysis also involves study of topological spaces, such as Frechet spaces, not endowed with a norm.

The important examples of study of vector spaces in functional analysis are Hilbert spaces and Banach spaces. Finite-dimensional Hilbert spaces are understood in linear algebra. They can be easily classified in functional analysis. While, the Banach spaces are little more complicated than the Hilbert spaces. Usually, they cannot be classified as simple as Hilbert.

Applications

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The subject of functional analysis plays a vital role in a number of fields. The areas of application of functional analysis include theory of functional spaces, theory of abstract spaces, representation theory, theory of operators, theory of operator equations, spectral theory and theory of normed rings. The functional analysis can be applied in engineering and technology, mechanics, quantum physics, theoretical physics etc.