Lecture

Functional Analysis I

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Description

This lecture introduces the concept of sublinear functionals in a real vector space, discussing their properties and applications. It covers the extension of linear functionals, Hahn-Banach theorem, and bounded linear functionals. The lecture also explores the proof of the extension of a linear functional and the properties of linear functionals in a vector space.

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Ontological neighbourhood

Mathematics

Analysis: Functional analysis

Related concepts (30)

Continuous linear operator

In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator. Continuous function (topology) and Discontinuous linear map Bounded operator Suppose that is a linear operator between two topological vector spaces (TVSs). The following are equivalent: is continuous.

Linear form

In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If V is a vector space over a field k, the set of all linear functionals from V to k is itself a vector space over k with addition and scalar multiplication defined pointwise. This space is called the dual space of V, or sometimes the algebraic dual space, when a topological dual space is also considered.

Sublinear function

In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space is a real-valued function with only some of the properties of a seminorm. Unlike seminorms, a sublinear function does not have to be nonnegative-valued and also does not have to be absolutely homogeneous. Seminorms are themselves abstractions of the more well known notion of norms, where a seminorm has all the defining properties of a norm that it is not required to map non-zero vectors to non-zero values.

Bounded set (topological vector space)

In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be inflated to include the set. A set that is not bounded is called unbounded. Bounded sets are a natural way to define locally convex polar topologies on the vector spaces in a dual pair, as the polar set of a bounded set is an absolutely convex and absorbing set. The concept was first introduced by John von Neumann and Andrey Kolmogorov in 1935.

Functional analysis

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. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous or unitary operators between function spaces.

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