Relationship of transpose of linear mapping and Riesz isomorphism

Question

How do I show the following:

Be $Z$ a limited Euclidean vector space with the inner product space $\langle \cdot,\cdot \rangle$ and $\Phi : Z \to Z^*, z \mapsto \langle \cdot,z \rangle$ (Riesz representation theorem). For $f \in \mathcal{L}(Z,Z)$ the following relationship holds:

$^tf = \Phi^{-1} \circ f^* \circ \Phi$

Answer 1

Let $x,y \in Z$. Then : \begin{align} \langle x, \Phi^{-1}\circ f^* \circ \Phi(y) \rangle &= f^*\circ\Phi(y)(x) \\ &= \Phi(y)\circ f(x)\\ &= \langle f(x) ,y \rangle \\ &= \langle x, f^t(y) \rangle \end{align}

Therefore $ \Phi^{-1}\circ f^* \circ \Phi = f^t$