Dual Space

• categories: Functional analysis, Definition

Dual Space The dual space of a vector space $V$ over a field $F$, denoted $V^{\ast }$ or $V^{\prime }$, is the set of all Linear Functionals from $V$ to $F$.

Definition

If $V$ is a vector space over a field $F$, then:

$$V^{\ast }=\{f:V\to F\mid f\text{ is linear}\}$$

Each element $f\in V^{\ast }$ is called a linear functional.

Properties

1. Vector Space Structure: $V^{\ast }$ is itself a vector space over $F$, where addition and scalar multiplication are defined pointwise:

   $$(f+g)(v)=f(v)+g(v),(\alpha f)(v)=\alpha \cdot f(v)$$

   for $f,g\in V^{\ast }$, $v\in V$, and $\alpha \in F$.

2. Dimension: If $V$ is finite-dimensional, $\dim (V^{\ast })=\dim (V)$. If $\{e_1,e_2,\dots ,e_n\}$ is a basis of $V$, then there exists a dual basis $\{e^1,e^2,\dots ,e^n\}$ in $V^{\ast }$ such that:

   $$e^i(e_j)={\delta }_{ij}$$

   where ${\delta }_{ij}$ is the Kronecker delta.

3. Evaluation Map: There exists a natural evaluation map $V\to V^{\ast \ast }$ (the double dual), defined by:

   $$v\mapsto \hat{v},\hat{v}(f)=f(v)\text{for }f\in V^{\ast }$$

   This map is an isomorphism if $V$ is finite-dimensional.

Example

For $V={\mathbb{R}}^n$, elements of $V^{\ast }$ can be represented as row vectors, acting on column vectors in $V$ by matrix multiplication.