Kernel of a Nonzero Functional Has Codimension One

Let $X$ be a vector space over $\mathbb{R}$ or $\mathbb{C}$, and let $f:X\to\mathbb{K}$ be a nonzero linear functional with kernel $\ker f$.

Proposition:

Context: The kernel of a nonzero functional is a codimension-one subspace, and translates of such kernels are precisely hyperplanes in real vector spaces; see hyperplanes as level sets .