Linear Functional

• categories: Functional analysis, Definition

A linear functional is a Linear map from a vector space $V$ over a field $F$ to the field $F$ itself

Intuition

A linear functional “evaluates” vectors by assigning them scalar values, preserving the structure of vector addition and scalar multiplication. For example, in $F^n$, a linear functional can often be represented by a dot product with a fixed vector

Properties

• The set of all linear functionals on $V$ forms the Dual Space $V^{\ast }$.
• If $V$ is finite-dimensional with basis $\{e_1,e_2,\dots ,e_n\}$, any linear functional $\phi \in V^{\ast }$ can be written as: $\phi (x)={\sum }_{i=1}^n{a}_i{x}_i$ where $a_i\in F$ are constants and $x_i$ are the coordinates of $x$ in this basis.

Example

For $V={\mathbb{R}}^n$, a linear functional $\phi :{\mathbb{R}}^n\to \mathbb{R}$ can be given by: $\phi (x)=a_1{x}_1+a_2{x}_2+\cdots +a_n{x}_n$ where $(a_1,a_2,\dots ,a_n)$ is a fixed vector in ${\mathbb{R}}^n$.