---- > [!definition] Definition. ([[perfect pairing]]) > Let $k$ be a [[field]] and $V,W$ be $k$-[[vector space|vector spaces]]. > A **perfect pairing** is a [[bilinear map]] $\langle -,- \rangle :V \times W \to k$ such that the maps $\begin{align} V &\to W^{*} \\ v & \mapsto \langle v, - \rangle \end{align}$ and $\begin{align} W &\to V^{*} \\ w & \mapsto \langle -, w \rangle \end{align}$ are both [[linear isomorphism|isomorphisms]]. ^definition > [!specialization] > In finite dimensions, one of the above maps being an [[isomorphism]] guarantees the other to be ([[double dual of a finite-dimensional vector space is naturally isomorphic to that space|by]]), and indeed in finite dimensions a pairing is perfect if and only if it is [[nondegenerate bilinear form|nondegenerate]]. ^specialization ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```