Dual space in the context of Abstract index notation

Multilinear algebra is the study of functions with multiple vector-valued arguments, with the functions being linear maps with respect to each argument. It involves concepts such as matrices, tensors, multivectors, systems of linear equations, higher-dimensional spaces, determinants, inner and outer products, and dual spaces. It is a mathematical tool used in engineering, machine learning, physics, and mathematics.

In mathematical analysis, a bump function (also called a test function) is a function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ on a Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump functions with domain ${\displaystyle \mathbb {R} ^{n}}$ forms a vector space, denoted ${\displaystyle \mathrm {C} _{0}^{\infty }(\mathbb {R} ^{n})}$ or ${\displaystyle \mathrm {C} _{\mathrm {c} }^{\infty }(\mathbb {R} ^{n}).}$ The dual space of this space endowed with a suitable topology is the space of distributions.

In mathematics, specifically linear algebra, a degenerate bilinear form ${\displaystyle f(x,y)}$ on a vector space ${\displaystyle V}$ is a bilinear form such that the map from ${\displaystyle V}$ to ${\displaystyle V^{*}}$ (the dual space of ${\displaystyle V}$) given by ${\displaystyle v\mapsto (x\mapsto f(x,v))}$ has a non-trivial kernel, i.e. there exist some non-zero ${\displaystyle x}$ in ${\displaystyle V}$ such that ${\displaystyle f(x,y)=0}$ for all ${\displaystyle y\in V}$.

An equivalent definition when ${\displaystyle V}$ is finite-dimensional is that the previous map is not an isomorphism.

This article is mainly concerned with the second concept, which arose in the early 18th century as part of the calculus of variations. The first concept, which is more modern and abstract, is discussed in detail in a separate article, under the name linear form. The third concept is detailed in the computer science article on higher-order functions.

In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. The theorem also shows that there are sufficient continuous linear functionals defined on every normed vector space in order to study the dual space. Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem, and has numerous uses in convex geometry.