Equipping a vector space with an inner product results in a natural isomorphism $\CV\to\CV^\ast$, where the metric tensor can be interpreted as the linear mapping $\Bg:\CV\to\CV^\ast$ and its inverse $\Bg\inv:\CV^\ast\to\CV$.

Notation: Given two real vector spaces $\CV$ and $\CW$, we denote their inner products as $\dabrn{\cdot,\cdot}_{\CV}$ and $\dabrn{\cdot,\cdot}_{\CW}$ respectively. Given vectors $\Bv\in\CV$ and $\Bw\in\CW$, we define their lengths as

Regarding $\CV$ and $\CW$,

1. their bases are denoted $\cbrn{\BE_A}$ and $\cbrn{\Be_a}$,
2. their dual bases are denoted $\cbrn{\BE^A}$ and $\cbrn{\Be^a}$,
3. their metrics are denoted $\BG$ and $\Bg$ with the components $G_{AB}=\dabrn{\BE_A,\BE_B}_\CV$ and $g_{ab}=\dabrn{\Be_a,\Be_b}_\CW$,

respectively. Here, the indices pertaining to $\CV$ are uppercase $(ABC\dots)$ and the indices pertaining to $\CW$ are lowercase $(abc\dots)$.

Definition: Let $\BP:\CV\to\CW$ be a linear mapping. Then the transpose, or adjoint of $\BP$, written $\BP\tra$, is the linear mapping

for all $\Bv\in\CV$ and $\Bw\in\CW$. Carrying out the products,

For arbitrary $\Bv$ and $\Bw$,

from which we can obtain the components of the transpose as

If $\BB:\CV\to\CV$ is a linear mapping, it is called symmetric if $\BB=\BB\tra$.

Definition: Let $\BP:\CV\to\CW$ be a linear mapping. Then the dual of $\BP$ is a metric independent mapping

defined through natural pairings for all $\Bv\in\CV$ and $\Bbeta\in\CW^\ast$. Carrying out the products,

For arbitrary $\Bv$ and $\Bbeta$, we obtain the components of the dual mapping as

To fully appreciate the symmetry that originates from the duality, we can think of not just the mappings between $\CV$ and $\CW$, but also between their dual spaces. To this end we can enumerate four mappings corresponding to $\cbr{\CV,\CV^\ast}\to\cbr{\CW,\CW^\ast}$ and their duals, corresponding to $\cbr{\CW,\CW^\ast}\to\cbr{\CV,\CV^\ast}$. Their definitions can be found in the table below.

 $\BP\in\CW \dyd\CV^\ast$ $P^a_{\idxsep A}=\BP(\Be^a,\BE_A)$ $\BP = P^{a}_{\idxsep A}\, \Be_a \dyd \BE^A$ Mappings Duals

The commutative diagrams pertaining to these mappings can be found in the figure below