Calculus in vector spaces without norm by Alfred Frölicher, W. Bucher

By Alfred Frölicher, W. Bucher

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V. a) = U q( ~. ~)" ~ II II V. 1BII. a~E 1 and the norm is continooos, II~'~IlJJR, hance also e r ( W,S)II ~ ~, whioh yields, by the definition of the pssodotopology induced by the norm: er( W , ~ ) ~ r& R(E1 ; E2) . E1 • This shows that - (2) 43 - Suppose conversely that r~ R(E1;E2) and let q be as before. It is sufficient to show that q is continuous at the point Oo So let ~ E l ; we shall show that then q(~)~IR. We still introduce the map s: El ~ E l by I s(x) = l . x for x ~ O , Itxlt 0 for x = O.

Vector space with its natural topology and E 2 a normed vector space, then C~(E1;E2) consists exactly of the continuous maps from E 1 into E 2. Proof. 3)) that, even for arbitrary El,E2, the elements of --oC(El;E2) are continuous. Let now~ conversely, a quasi-bounded filter~on set. Its closure, f: E1---JPE2 be continuous. E 1 . 1), ~ c o n t a i n s Consider a bounded which we denote by A, is then a set of ~ is closed and bounded. which BuG such a set in a finite dimensional vector space is comp~act, f being continuous, f(A) is compact.

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. 2) (FlVr2)( ) = rl( ) which is a consequence of the set-theoretic equality (FIWF2)(B) = FI(B) w F2(B).

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