Dual Vector Spaces and Double Duals

Dual Vector Spaces

• With $V$ a vector space over field $F$, the dual vector space of $V$ is defined as $V^* = {f: V \rightarrow F \ | \ f \text{ is linear} }$.
• For $V = \mathbb{R}^n$, any $f \in (\mathbb{R}^n)^*$ can be expressed as $f(x_1, \dots, x_n) = a_1 x_1 + \dots + a_n x_n$, where $a_i \in \mathbb{R}$.
• $(\mathbb{R}^n)^* = {a_1 x_1 + \dots + a_n x_n \ | \ a_i \in \mathbb{R} }$

Basis of $V^*$

• For a finite-dimensional vector space $V$ with basis $B = {v_1, \dots, v_n }$,
• Define $f_i: V \rightarrow F$ where $f_i(v_j) = 1$ if $i = j$, and $f_i(v_j) = 0$ if $i \neq j$.
• ${f_1, \dots, f_n }$ is a basis of $V^*$, known as the dual basis of $B$.
• If $V$ is finite-dimensional, the dimensions of $V^$ and $V$ are equal: $\dim(V^) = \dim(V)$.

Double Dual

• The double dual space $V^{}$ is defined as $V^{} = (V^)^ = {g: V^* \rightarrow F \ | \ g \text{ is linear} }$.
• A mapping $\phi: V \rightarrow V^{**}$ is defined such that $\phi(v)(f) = f(v)$ for all $f \in V^*$.
• $\phi$ establishes an isomorphism, indicating $V \cong V^{}$, allowing for the identification of $V$ and $V^{}$.