Let X be a normed space over the scalar field of either all real numbers or of all complex numbers.
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.
Date and Venue
Start Date
Venue
Room 0.03, DMP
Speaker
B.V. Limaye
Area
General seminar of CMUP