The transpose of a linear injection is surjective. Let 
$$T:V\longrightarrow W$$
be a linear map (of vector spaces), and let
\begin{eqnarray}
T^*:W^* &\longrightarrow& V^* \\
f\ &\longmapsto& f\circ T
\end{eqnarray}
be its transpose (or dual map). 
How to show that if $T$ is injective then $T^*$ must be surjective, and that if $T$ is surjective then $T^*$ must be injective?

If $T$ is injective then it has a inverse $T^{-1}:T(V)\rightarrow V$. Take $v\in V^\star$ and define $f\in W^\star$ by $f(x)=v(T^{-1}(x))$. Then $T^\star(f)=f\circ T=v\circ T^{-1}\circ T=v$. Hence,$T^{*}$ is surjective.
On the other hand, suppose that  $T$ is surjective. If $T^\star f=T^\star g$, then $f(T(x))=g(T(x))$ for all $x\in V$. Hence $(f-g)(T(x))=0$ for all $x\in V$. Because $T$ is surjective you can conclude that $f-g=0$, or $f=g$. 
