# Definition Give two [[Inner Product Space|inner product spaces]] $V, W$ equipped with [[Isomorphism|isomorphisms]] $g:V\rightarrow V^*$ and $G:W\rightarrow W^*$, assume we have a [[Linear Map|linear map]] $f:V \rightarrow W$, then we may define the adjoint of $f$, $\tilde{f}$, as: $ G(\pmb{w}, f \pmb{v}) = g(v, \tilde{f} \pmb{w}) $ where $\pmb{v} \in V, \pmb{w} \in W$. That is, the adjoint is a map $\tilde{f}:W \rightarrow V$. See the [[Toy Index Thoerem]] for an important relationship between the [[Kernel of a Map|kernels]] of $f$ and $\tilde{f}$.