I was wondering if the in the title mentioned statement is true or false? I thought I read about it but I'm not sure if it's true and provable or not.
So if we have $V$ a K-vector space and $U \subset V$ a subspace. We take $\phi$: $U \to V$ as the obvious inclusion map. Is the transpose map $\phi^T: V^* \to U^*$ surjective?