Linear Functional on Vector Space is Zero or Surjective

Theorem
Let $K$ be a field.

Let $X$ be a vector space over $K$.

Let $f : X \to K$ be a linear functional.

Then either:


 * $\map f x = 0$ for each $x \in X$

or:


 * $f$ is surjective.

Proof
Suppose that $\map f {x_0} \ne 0$ for $x_0 \in X$.

Take $c \in K$.

Then we have, from linearity:

Since $c \in K$ was arbitrary, we have that $f$ is surjective.