Proposition 5

Let W be a weakly closed vector subspace of $L^{\infty}(G)$ invariant under translation.

a) For every neighbourhood U of A(W) in $\widehat{G}$, every function of W is the weak limit of linear combinations of characters belonging to U;

b) If the boundary of A(W) contains no nonempty perfect set, every function of W is the weak limit of linear combinations of characters belonging to W.

To prove a), it is enough by the bipolar theorem to show that if $f$ is a function of $L^1(G)$ orthogonal to the elements of U, then $f$ is orthogonal to W. Now the Fourier cotransform $\overline{\mathscr{F}}(f)$ then vanishes on the neighbourhood U of $A(W) = V(W^{\circ})$, so that prop. 2 of II, p. 252 shows indeed that $f\in W^{\circ}$. Assertion b) is established similarly, using th. 2 of II, p. 257 instead of prop. 2 of II, p. 252.