The nullspace of a linear functional that is not $\equiv 0$ is a linear subspace of codimension $1$.
I don't understand this statement on page 57, Functional Analysis(Pater Lax). Does it mean the dimension of nullspace of a linear functional is either zero or the dimension of the domain of the functional minus one, which I don't see why it's necessarily true.
Added: Thank you all for your valuable comments and answers. I didn't realize that it's wrong to interpret "The nullspace of a linear functional that is not $\equiv 0$" as the nullspace (of a linear functional) that is not $\equiv 0$ until I saw the answers.