Why does there exists a self-adjoint operator $T$?

Question

In an inner product space $V$ with inner product $\langle\cdot,\cdot\rangle$ with a second inner product $g$ on $V$ , why does there exists an self-adjoint operator such that

$$\langle x,y\rangle= g(T(x),y)$$

for all $x,y\in V$?

Answer 1

To make the problem a bit more symmetric, suppose that $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$ are inner products on the same finite-dimensional space $V$. Choose an orthonormal basis $\{ e_n \}_{n=1}^{N}$ of $V$ with respect to the second inner product. Then $\langle e_n,e_{n'}\rangle_2 = \delta_{n,n'}$, and \begin{align} \langle x,y\rangle_1 & = \left\langle x, \sum_{n=1}^{N}\langle y,e_n\rangle_2 e_n\right\rangle_1 \\ & = \sum_{n=1}^{N}\langle e_n,y\rangle_2 \langle x,e_n\rangle_1 \\ & = \sum_{n=1}^{N}\langle x,e_n\rangle_1 \langle e_n,y\rangle_2 \\ & = \left\langle \sum_{n=1}^{N}\langle x,e_n\rangle_1 e_n,y \right\rangle_2 \end{align} So, the linear operator $T : V\rightarrow V$ defined by $$ Tx = \sum_{n=1}^{N}\langle x,e_n\rangle_1 e_n. $$ satisfies $$ \langle x,y\rangle_1 = \langle Tx,y\rangle_2 $$