Vector space/Finite-dimensional/Inner product/Adjoint endomorphism/Existence/Fact/Proof

Proof

Let

$\phi :V⟶V$

be given, and let $w\in V$ be fixed. Then, the mapping

$V⟶𝕂,v⟼⟨\phi (v),w⟩,$

is a linear form on $V$. Therefore, there exists (due to fact in the real case; for the complex case see exercise) a right gradient $r=:\hat{\phi }(w)$ in $V$ (uniquely determined by $\phi$ and $w$) fulfilling

$⟨v,\hat{\phi }(w)⟩=⟨\phi (v),w⟩.$

We have to show that the assignment

$w⟼\hat{\phi }(w)$

is linear. We have

$\begin{array}{cc}⟨v,\hat{\phi }(w_1+w_2)⟩& =⟨\phi (v),w_1+w_2⟩\\ & =⟨\phi (v),w_1⟩+⟨\phi (v),w_2⟩\\ & =⟨v,\hat{\phi }(w_1)⟩+⟨v,\hat{\phi }(w_2)⟩\\ & =⟨v,\hat{\phi }(w_1)+\hat{\phi }(w_2)⟩.\end{array}$

As this holds for all $v\in V$, we have

$\hat{\phi }(w_1+w_2)=\hat{\phi }(w_1)+\hat{\phi }(w_2).$

Moreover,

$\begin{array}{cc}⟨v,\hat{\phi }(sw)⟩& =⟨\phi (v),sw⟩\\ & =\bar{s}⟨\phi (v),w⟩\\ & =\bar{s}⟨v,\hat{\phi }(w)⟩\\ & =⟨v,s\hat{\phi }(w)⟩.\end{array}$

As this holds for all $v\in V$, we get

$\hat{\phi }(sw)=s\hat{\phi }(w).$

To fact