A linear topological space X over the real field R is said to be a paranormed space if there is a subadditive function
g:X [member of] R such that g([theta]) = 0, g(x) = g(-x), and scalar multiplication is continuous, i.e., [absolute value of [[alpha].sub.n] - [alpha]] [right arrow] 0 and g([x.sub.n] - x) [right arrow] 0 imply g([[alpha].sub.n][x.sub.n] - ax) [right arrow] 0 for all x's in X and a's in R, where [theta] is the zero vector in the linear space X.
First, we use a more general term on the right hand side of (1.2), namely a+[theta] [[psi]([gamma](x)) + [psi]([gamma](y))], where a and [theta] are positive constants, [gamma] : G [right arrow] (0, [infinity]) is a function satisfying some special conditions to be discussed in the next section, and [psi] : [0, [infinity]) [right arrow] (0, [infinity]) is an increasing subadditive function
. Second, we replace the domain of the function f by some of noncommutative group G.
because the maximum over y is a subadditive function
and is invariant by translations of y by a fixed value.
Brown's proof (1960) where these roots are replaced with a nonnegative nondecreasing subadditive function
f defined on (0, [infinity]).
Aydi , in correcting the proof of , restricts the theorem to the class of subadditive functions
[phi]; i.e., those for which inequality (1.4) is true.