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Let V be a vector space over field F, is there always a linear functional T: V → F such that T is injective? I know this is equivalent to N(T)=0. If yes, how do you show such a T always exist?

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If $\dim V\geq 2$, let $\{e_{1},e_{2}\}$ be a linearly independent set and assume that $\varphi$ is injective, let $\varphi(e_{1})=k_{1}$ and $\varphi(e_{2})=k_{2}$ where $k_{1},k_{2}\ne 0$. So some $c$ is such that $k_{2}=ck_{1}$ and hence $\varphi(ce_{1}-e_{2})=0$. Injectivity implies $ce_{1}-e_{2}=0$, absurb.

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