By Alfred Frölicher, W. Bucher
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Additional info for Calculus in vector spaces without norm
E 2 which is linear is the zero map. Let x ~ E 1 . 2), ~V. [x]~E1, hence But since for a linear map r one has that ~r~,x) @ r ( ~ V , [ x ] ) ~ E 2. = r(x), it follows ~r( IV, Ix] ) = r(~x]) = ~r(x)] . So we have ~r(x)] ~ E2, hence r(x) = O. 2. Differentiability at a point. 8) we assume henceforth that all spaces El~ E2,... are separated. Let f: E1----~ E 2 be any map of pseudo-topological vector spaces, and a ~ E1 . 1) Proposition. There exists at most one ~eL(E1;E2) such that the map r defined by f(a+h) = f(a) + ~(h) + r(h) is a remainder.
1). 1). - b) Let 49 - Jl f be differentiable. o II fi' where i~l x J J i~l the projection map 7. 1). J - 65 - § 6. PSEUDO-TOPOLOGIES ON SOmE FUNCTION SPACES. 1. The spaces B(E1;E2), Co(El;E2) and L(EI;E2). (fl+f2)(~) & W . f2(@) (cf. f I + ~2~f2 of two quasi-bounded maps fl' f2 is also quasi-bounded. B~El= ~ ~(I~)~E 2. @e claim that this definition yields a compatible pseudotopology on the vector space _B(E1;E2). 2) hold. 1), only the second one, which demands that ~l v ~2~B(EI;E2) if ri ~B(EI;E2) for i = 1,2, is not obvious.
24 - For any pseudo-topological vector space E, the space E° defined above is a locally convex topological vector space. 10) E ~ E° , with equality if and only if E is itself a topological locally convex vector space. 8. Equabl~ continuity. 1) ~f(a,h) = f(a + h) - f(a). 2) Definition. 3) f: E1 ~ E 2 is called equably c o n t i ~ iff J Proposition. e. continuous at each point a ~ El). Proof. get Let ~ a El" Then, since A f( Ca] , ~ _ Fa] ) vI E 2. [a] $ E 1 and ~-[a] ~ E l, we But since Z~f(b,x-b) = f(x)-f(b) 4 f ( #a] , X -[a])~ f(~) - f([a]).