Dual Space

Definition: Dual Space

The dual space of a vector space $(V,F,+,\cdot )$ is a vector space $(V^{\ast },F,+,\cdot )$, where $V^{\ast }$ the set of all linear forms of $(V,F,+,\cdot )$ and the two operations $+:V^{\ast }\times V^{\ast }\to V^{\ast }$ and $\cdot :F\times V^{\ast }\to V^{\ast }$ are defined as

$$\begin{array}{rl}(\phi +\psi )(x)& =\phi (x)+\psi (x)\forall \phi ,\psi \in V^{\ast },\forall x\in V\\ (a\phi )(x)& =a\phi (x)\forall \phi \in V^{\ast },\forall x\in V\end{array}$$

NOTATION

$$V^{\ast }\mathrm{hom}(V,F)$$

Definition: Covector

The elements of the dual space of $V$ are known as its covectors.