1968/04/01 by I. J. Maddox
paper · doi:10.1017/s0305004100042894
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, l ;( p ) is the set of all complex sequences x = ( x k ) such that and m ( p ) is the set of x such that sup Throughout, sums and suprema without limits are taken from 1 to ∞. Simons (3) considered only the case in which 0 p k ≤ 1 so that natural paranorms would seem to be in m(p) . In fact Simons showed that g 1 was a paranorm for l ( p ), but that g 2 did not satisfy the continuity of multiplication axiom.