nLab linear functional

Definition

In linear algebra and functional analysis, a linear functional (often just functional for short) is a function $V \to k$ from a vector space to the ground field $k$. (This is a functional in the sense of higher-order logic if the elements of $V$ are themselves functions.) Then a linear functional is a linear such function, that is a morphism $V \to k$ in $k$-Vect. In the case that $V$ is a topological vector space, a continuous linear functional is a continuous such map (and so a morphism in the category TVS). When $V$ is a Banach space, we speak of bounded linear functionals, which are the same as the continuous ones. In an algebraic context, one may also use the term linear form, especially to distinguish from bilinear forms, quadratic forms, and the like.

Remarks

In a sense, linear functionals are co-probes for vector spaces. If the vector space $V$ in question has finite dimension and is equipped with a basis, then all linear functionals are linear combinations of the coordinate projections. These projections comprise the dual basis.

In infinite-dimensional topological vector spaces, the notion of dual basis breaks down once spaces more general than Hilbert spaces are considered. But for locally convex spaces, the Hahn–Banach theorem ensures the existence of ‘enough’ continuous linear functionals. Among non-LCSes, however, there are examples such that the only continuous linear functional is the constant map onto $0 \in k$.

Examples

• distribution

Related concepts

• nonlinear functional