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Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner product, norm, or topology) and the linear functions defined on these spaces and suitably respecting these structures.
However, while the results in real analysis are stated for real numbers, many of these results can be generalized to other mathematical objects. In particular, many ideas in functional analysis and operator theory generalize properties of the real numbers – such generalizations include the theories of Riesz spaces and positive operators .
In functional analysis and related fields, it refers to a mapping from a space into the field of real or complex numbers. [2] [3] In functional analysis, the term linear functional is a synonym of linear form; [3] [4] [5] that is, it is a scalar-valued linear map.
An Introduction to Complex Analysis in Several Variables. Van Nostrand. Rudin, Walter (1976). Principles of Mathematical Analysis. Walter Rudin Student Series in Advanced Mathematics (3rd ed.). McGraw-Hill. ISBN 9780070542358. Rudin, Walter (1986). Real and Complex Analysis (International Series in Pure and Applied Mathematics). McGraw-Hill.
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear operators acting upon these spaces and respecting these structures in a suitable sense.
5 Real and complex algebras. 6 Topological vector spaces. 7 Amenability. ... This is a list of functional analysis topics. See also: Glossary of functional analysis ...
Complex analysis – studies the extension of real analysis to include complex numbers; Functional analysis – studies vector spaces endowed with limit-related structures and the linear operators acting upon these spaces; Nonstandard analysis – studies mathematical analysis using a rigorous treatment of infinitesimals.
Every norm, seminorm, and real linear functional is a sublinear function.The identity function on := is an example of a sublinear function (in fact, it is even a linear functional) that is neither positive nor a seminorm; the same is true of this map's negation . [5] More generally, for any real , the map ,: {is a sublinear function on := and moreover, every sublinear function : is of this ...