# Proof: If the dual map $f^*$ is the null map, then $f$ is the null map

Let $V$ and $W$ be vector spaces, and let $f:V\to W$ be a linear map. Let $f^*:W^*\to V^*$ be the dual map ($f^*=f^T= f$ transposed). Prove that if $f^*$ is the null map, then $f$ is the null map.

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If $f^\ast$ is the null map, then for any $\varphi:W\to K$ we have that $\varphi f:V\to K$ is the null map. This means that ${\rm im \;}f\subseteq\ker \varphi$ for every $\varphi\in W^\ast$. What is $$\bigcap_{\varphi\in W^\ast}\ker\varphi\;?$$