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:

\(\ds \map f {c \paren {\map f {x_0} }^{-1} x_0}\)

\(=\)

\(\ds c \paren {\map f {x_0} }^{-1} \map f {x_0}\)

\(\ds \)

\(=\)

\(\ds c\)

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

$\blacksquare$