Let $\left\langle x,y\right\rangle$ be an inner product (in a real space). Then, there is a matrix $S$ symmetric s.t. $$\left\langle x,y\right\rangle=x^t Sy.$$
I want to have the form of the adjoint of $A$, i.e. the matrix $B$ s.t. $$\left\langle Ax,y\right\rangle=\left\langle x,By\right\rangle.$$
I did as follow, $$\left\langle Ax,y\right\rangle=(Ax)^tS y=x^tA^tSy=x^t\left(S^tA\right)^t y\underset{G\,\mathrm{sym}} {=}x^t\left(SA^t\right)^ty.$$
How can I get something like $x^t S By$ ? Bye the way, does my computation will prove the existence or not ?
You were pretty close. $S$ will not only be symmetric, but also positive deinite, this means that it will also be invertible.
So the two following expressions are equivalent:
$$\left\langle Ax,y\right\rangle=x^tA^tSy=x^tSBy=\left\langle x,By\right\rangle$$
$$A^tS= SB $$
And as $S$ is invertible.
$$ B = S^{-1}A^tS$$