Let X be a normed space over the scalar field of either all real numbers or of all complex numbers. Let p be a seminorm on X, that is:
(i) p(x)?0 for all x in X
(ii) p(x)+p(y)?p(x+y) for all x,y in X
(iii) p(kx)=|k|p(x) for all x in X and all scalars k.
It is easy to see that if p is continuous, then p is countably subadditive. In fact, if p is only lower semicontinuous, then p is countably subadditive. In 1936, Gelfand proved that if X is a Banach space and p is lower semicontinuous, then p is continuous. Generalizing this result, Zabreiko proved in 1969 that if X is a Banach space and p is countably subadditive, then p is continuous. Apparently this result of Zabreiko has remained largely unnoticed. It can be used to deduce several major theorems in Functional Analysis very easily.