Some discontinuous translation-invariant linear forms
The author proved in  that every translation-invariant linear form on (Rn), as well as on other spaces of test functions and distributions, is necessarily continuous. The same result has also been proved for the Hilbert space L2(G) where G is a compact connected Abelian group. In contrast to this it is proved here that there do exist discontinuous translation-invariant linear forms on the Banach spaces l1(Z) and L1(R), and on the Hibert spaces L2(D) and L2(R). Here Z denotes the additive group of the integers, D denotes the totally disconnected compact Abelian Cantor discontinuum group, and R denotes the additive group of the real numbers. The proofs divide into two parts: A general criterion (Theorem 1) and proofs that the spaces l1(Z), L2(D), L2(R), and L1(R) satisfy this criterion (Theorems 2, 3, 4, and 5, respectively).