Nonzero linear transposition

Question

Let $U,V$ be vector spaces and $U^\intercal, V^\intercal$ their duals. Let $T \in \operatorname{Hom}(V,U)$(A linear map) Denote by $T^\intercal \in \operatorname{Hom}(U^\intercal,V^\intercal)$ the dual of $T$. It is well-known that

$T = 0 \implies T^\intercal = 0$

When $U$ is finite-dimensional, the reverse implication holds, but what if $U$ is infinite-dimensional? Does there exist a $T \neq 0$ so that $T^\intercal = 0$?

Answer 1

If $\;0\neq T\in\text{Hom}\,(V,U)\;$ then there exists $\;0\neq v\in V\;$ such that $\;Tv=w\neq0\;$ , and then there always exists $\;f\in \text{Hom}\,(U,V)\;$ with $\;fw\neq0\;$ , so that

$$T'(f)(v):=f\circ T(v)=f(Tv)=fw\neq 0\implies T'\neq 0$$

and no matter what the dimensions are.