Dual vector space

A dual vector space $V^{*}$, or just dual space, is a special kind of Vector space associated with another vector space $V$. It is the space of all linear functionals on $V$, each defined as a linear map $a:V\to K$, where $K$ is the field both $V$ and $V^{*}$ are defined over. In symbols, the dual space is defined as:

The dimension of $V^{*}$ is the same as the dimension of $V$.

Being a vector space, the dual space comes equipped with a sum and a scalar multiplication operation:

• given $a,b\in V^{*}$, there exists a sum $(a+b)(v)=a(v)+b(v)$ for all $v\in V$.
• given $\alpha \in K$, there exists a scalar multiplication $(\alpha a)(v)=\alpha\cdot a(v)$ for all $v\in V$.

For any given Basis $\{ e_{1},\ldots,e_{n} \}$ in $V$, there exists a corresponding dual basis $\{ e_{1}^{*},\ldots,e_{n}^{*} \}$ composed of linear functionals that, when applied to the normal basis vectors, return the Kronecker delta: $e^{*}_{i}(e_{j})=\delta_{ij}$.