= Dual pairing
{title2=$\langle\varphi,v\rangle=\varphi(v)$}
= Evaluation pairing
{synonym}
The <dual pairing> evaluates a <linear functional> on a <vector>:
$$
\langle\varphi,v\rangle=\varphi(v),\qquad (\varphi,v)\in V^*\times V.
$$
It is <bilinear>: it is linear in each argument separately. Even over the <complex numbers>, this algebraic evaluation has no <complex conjugation>. An <inner product> instead has two <vectors> as inputs and, over the <complex numbers>, is <sesquilinear>. For $\varphi(x,y)=2x-y$ and $v=(3,1)$, the <dual pairing> equals five. Its value is independent of the <basis> used to describe its two inputs.
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