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

By Alfred Frölicher, W. Bucher

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28 - We denote by L (£1;£2) the vector space formed by the l i n e a r continuous maps from E1 to E2. 3), as in the topological case, the continuity of a linear map at zero implies the continuity at each point. Howevsr~ for bilinsar (and multilinsar) maps, the situa- tion is different: continuity at the origin does not necessarily imply continuity at all points. 8) Lemma. 9) we use that for a bilinear map one has ~b((al,a2),(hl,h2)) = b(al,h 2) + b(hl,a 2) + b(hl,h2). (1) In order to prove first that the given conditions are necessary, suppose b equably continuous and let Ill ~£1; |V A2 ~ £2" We put ~ = [0] x J~2; ]( = ll x GO].

U1 ~ ~l This means: for every U 2 ~ ~ 2 there exists such that f(y) - f(z)G U 2 for all y, z • A + V with y-z • UI. :Z3f(A,U1)¢U 2. Therefore U 2 ~ z ~ f ( J ~ , ~ l ) . e. ~ f ( J ~ , X ) ~ E2. This establishes the equable continuity of f. Proposition. 7) the origin, then ~ is quasi-bounded and equably continuous, Proof. x) and ~ (a,h) = ~ (h), get . and = and hence the continuity at the point zero yields the two assertions. - 28 - We denote by L (£1;£2) the vector space formed by the l i n e a r continuous maps from E1 to E2.

I) = %. t (i) = ~ . a . Hence r(~) Since q(~) -- ¢ ( , ~ . a - a. 2)@r(~V,1)~, E, hence - a~ E resp. q( IV)~a E, and that means exactly that lim q(~) = a ; the proof is complete. ~--~o In the classical case, the converse of this proposition holds. However it seems, that in the general case, our differentiability condition is a little bit stronger. Again some question of equability comes in. 3) Pro~osltion. 2) is sufficient for the differentiability of f: IR the point ~ . (*) more generally if E is admissible (see appendix (i)).