Throughout $\phi$ be a continuous character from a locally compact abelian group $G$ to the circle.
I'm trying to understand this implication. Basically we want to show that a certain representation is irreducible. We know that the representation is of the form $C_c(G)/W$, where $W$ is the subspace of functions $f$ which are degenerate with respect to the norm $\langle\cdot\rangle_\phi$. We have that $\langle f,f\rangle_\phi=\left|\int\overline{\phi(s)}f(s)\ ds\right|^2$, and the authors assert that hence the subspace degenerate with respect to $\langle\cdot\rangle_\phi$ has codimension one, and therefore $C_c(G)/W$ (more precisely its closure) is one-dimensional.
How do we know that the degenerate subspace has codimension one? How does it follow from the calculation of the norm of an arbitrary $f\in C_c(G)$?