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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 $f\in C_c(G)$?

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up vote 2 down vote accepted

The degenerate subspace is the orthogonal complement of $\mathbb{C}\phi$ w.r.t. the inner product $\int\bar f g ds$. Whence codimension $1$.

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