TITLE: Does a non-zero linear functional attain every value?
QUESTION [2 upvotes]: I want to prove the following 

If $y$ is a non-zero linear functional on a vector space $\mathbb V$, and if $\alpha$ is an arbitrary scalar, does there neccessarily exist a vector $x \in \mathbb V$ such that $[x,y] = \alpha$ ?

EDIT: Removed my attempt at it.

REPLY [10 votes]: If $y$ is nonzero, then there is a $v_0 \in V$ such that $y(v_0)=r\neq 0$. And if $\alpha \in \mathbb R$, then $y({\alpha \over r}v_0)={\alpha \over r}y(v_0)=\alpha$. So $y$ is surjective.
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