- Mathematical Proceedings of the Cambridge Philosophical Society / Volume 64 / Issue 02 / April 1968, pp 335-340
- Copyright © Cambridge Philosophical Society 1968
- DOI: http://dx.doi.org/10.1017/S0305004100042894 (About DOI), Published online: 24 October 2008

^{a1 }University of Lancaster

Abstract

A paranormed space *X* = (*X*, *g*) is a topological linear space in which the topology is given by paranorm *g*—*a* real subadditive function on *X* such that *g*(θ) = 0, *g*(*x*) = *g*(−*x*) and such that multiplication is continuous. In the above, θ is the zero in the complex linear space *X* and continuity of multiplication means that λ_{n} → λ, *x*_{n} → *x*(i.e. *g*(*x*_{n} − *x*) → 0) imply λ_{n}*x*_{n} → λ*x*, for scalars λ and vectors *x*. We shall use the term semimetric function to describe a real subadditive function *g* on *X* such that *g*(^{θ}) = 0, *g*(*x*) = *g*(−*x*). Two familiar paranormed sequence spaces, which have been extensively studied (3), are *l*(*p*) and *m*(*p*). For a given sequence *p* = (*g _{k}*) of strictly positive numbers,

(Received May 09 1967)