Inner product/Adjoint endomorphism/Definition

Let ${\displaystyle {}V}$ be a ${\displaystyle {}{\mathbb {K} }}$-vector space, endowed with an inner product ${\displaystyle {}\left\langle -,-\right\rangle }$, and let

${\displaystyle \varphi \colon V\longrightarrow V}$

denate an endomorphism. An endomorphism

${\displaystyle \psi \colon V\longrightarrow V}$

is called adjoint to ${\displaystyle {}\varphi }$, if

${\displaystyle {}\left\langle \varphi (v),w\right\rangle =\left\langle v,\psi (w)\right\rangle \,}$

holds for all ${\displaystyle {}v,w\in V}$.