Translation-invariant linear forms on p-adic continuous functions vanish
= Translation-invariant linear forms on p-adic continuous functions vanish
{title2=$T=0$}
Let $T:\mathcal C(\mathbb Z_p,\mathbb Q_p)\to\mathbb Q_p$ be linear and invariant under translation by one. For any $f$, choose its continuous <discrete antiderivative> $Jf$. Then $T(f)=T(\Delta Jf)=T(Jf(\cdot+1))-T(Jf)=0$. No <continuity> or boundedness of $T$ is assumed. In particular this applies to a form invariant under every translation by a <p-adic integer>.