TITLE: Dual space and linear functional
QUESTION [5 upvotes]: The problem is this: Let $V$ be a vector space over $\mathbb{K}$ and $v \in V$. Show that if $f(v) = 0, \forall$ $f \in L(V, \mathbb{K})$, then $v=0$.
It's a problem of a book I'm using to study Dual Spaces. It seems easy to solve, but I'm out of ideas... I tried write $v$ as a linear combination of the basis of $V$ and $f$ as a combination of the dual basis, but then I realized dimension of $V$ can be infinity.

REPLY [6 votes]: Assume $v\ne0$ so let $V'$ such that
$$V=V'\oplus\operatorname{span}(v)$$ 
and let $g$ the linear form such that $g(v)=1$ and $g(x)=0,\;\forall x\in V'$ and this contradicts that $f(v)=0,\;\forall f\in L(V,\Bbb K)$.