Vector space/Inner product/Endomorphism/Sesquilinear form/Fact

Let ${}V$ be a ${}{\mathbb {K} }$-vector space, endowed with an inner product. Then the following statements hold.

1. The assignment

   $\varphi \longmapsto \Psi _{\varphi }$

   assigns to an endomorphism a sesquilinear form. Hence,

   $\operatorname {End} _{}{\left(V\right)}\longrightarrow \operatorname {Sesq} _{}{\left(V\right)},\varphi \longmapsto \Psi _{\varphi }.$

2. This assignment is linear; it is bijective if ${}V$ has finite dimension.
3. Let ${}V$ be finite-dimensional. The endomorphism ${}\varphi$ is bijective if and only if ${}\Psi _{\varphi }$ is not degenerate.
4. Let ${}V$ be finite-dimensional. The endomorphism ${}\varphi$ is self-adjoint if and only if ${}\Psi _{\varphi }$ is Hermitian.

Proof, Write another proof