Is that true that for every linear transformation $\phi : V^* \to W^*$ there is a linear transformation $\psi: W \to V$ such that $\psi^* = \phi$?

Question

$V$ and W are finitely dimensional linear spaces over the field $K$. Is that true that for every linear transformation $\phi : V^* \to W^*$ there is a linear transformation $\psi: W \to V$ such that $\psi^* = \phi$?

• W* means dual space of W, that is a space of all functionals $W \to K$
• V* means dual space of V, that is a space of all functionals $V \to K$

It is like an invert definition of dual mapping. It must be something simple, I just can't think of it anymore.

Answer 1

Yes, it is true. You can consider the transpose map $\phi^*\colon W^{**}\to V^{**}$ and the canonical isomorphisms $\omega_W\colon W\to W^{**}$, $\omega_V\colon V\to V^{**}$.

Set $\psi=\omega_V^{-1}\circ\phi^*\circ\omega_W$ and finish up.