Dual vector spaces

We have a \(n \in \mathbb{N}\) finite dimensional vector space \(V\) such that \(\dim V = n\), with a basis \(\{\mathbf{e}_i\}_{i=1}^n\) and a field \(F\). In the following sections we make use of the Einstein summation convention introduced.

Definition 1: let \(\mathbf{\hat f}: V \to F\) be a covector or linear functional on \(V\) if for all \(\mathbf{v}_{1,2} \in V\) and \(\lambda, \mu \in F\) we have

\[ \mathbf{\hat f}(\lambda \mathbf{v}_1 + \mu \mathbf{v}_2) = \lambda \mathbf{\hat f}(\mathbf{v}_1) + \mu \mathbf{\hat f}(\mathbf{v}_2). \]

Throughout this section covectors will be denoted by hats to increase clarity.

Definition 2: let the the dual space \(V^* \overset{\text{def}} = \mathscr{L}(V, F)\) denote the vector space of covectors on the vector space \(V\).

Each basis \(\{\mathbf{e}_i\}\) of \(V\) therefore induces a basis \(\{\mathbf{\hat e}^i\}\) of \(V^*\) by

\[ \mathbf{\hat e}^i(\mathbf{v}) = v^i, \]

for all \(\mathbf{v} = v^i \mathbf{e}_i \in V\).

Theorem 1: the dual basis \(\{\mathbf{\hat e}^i\}\) of \(V^*\) is uniquely determined by

\[ \mathbf{\hat e}^i(\mathbf{e}_j) = \delta_j^i, \]

for each basis \(\{\mathbf{e}_i\}\) of \(V\).

Proof:

Let \(\mathbf{\hat f} = f_i \mathbf{\hat e}^i \in V^*\) and let \(\mathbf{v} = v^i \mathbf{e}_i \in V\), then we have

\[ \mathbf{\hat f}(\mathbf{v}) = \mathbf{\hat f}(v^i \mathbf{e}_i) = \mathbf{\hat f}(\mathbf{e}_i) v^i = \mathbf{\hat f}(\mathbf{e}_i) \mathbf{\hat e}^i(\mathbf{v}) = f_i \mathbf{\hat e}^i (\mathbf{v}), \]

therefore \(\{\mathbf{\hat e}^i\}\) spans \(V^*\).

Suppose \(\mathbf{\hat e}^i(\mathbf{e}_j) = \delta_j^i\) and \(\lambda_i \mathbf{\hat e}^i = \mathbf{0} \in V^*\), then

\[ \lambda_i = \lambda_j \delta_i^j = \lambda_j \mathbf{\hat e}^j(\mathbf{e}_i) = (\lambda_j \mathbf{\hat e}^j)(\mathbf{e}_i) = \mathbf{0}, \]

for all \(i \in \mathbb{N}[i \leq n]\). Showing that \(\{\mathbf{\hat e}^i\}\) is a linearly independent set.

Obtaining a vector and consequent covector space having the same dimension \(n\).

From theorem 1 it follows that for each covector basis \(\{\mathbf{\hat e}^i\}\) of \(V^*\) and each \(\mathbf{\hat f} \in V^*\) there exists a unique collection of numbers \(\{f_i\}\) such that \(\mathbf{\hat f} = f_i \mathbf{\hat e}^i\).

Theorem 2: the dual of the covector space \((V^*)^* \overset{\text{def}} = V^{**}\) is isomorphic to \(V\).

Proof:

Will be added later.