By Alfred Frölicher, W. Bucher

ISBN-10: 3540036121

ISBN-13: 9783540036128

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**Extra info for Calculus in vector spaces without norm**

**Example text**

A) b) Proof. holds since ~0] ~ E implies ~ ~[0] and . 3). b) Same argument, using that ~V. Ix] ~ E. c) For ~ = O, this is a). V is convex. - d) Let V e ~ ' . 4), V cnntains a set which is convex and satisfies U = [0,~ . U. By c), V ~ V n(-V) ~ U ~ (-U) ( ~ I~I ~ l 23 and x e U since U E I~ c~ . Thus we have . The set ~/= U ~ ( - U ) = ~/. In fact, if z 6 I l . e. x ( U and - x ( U o is convex , then z = ~x, where Since ~ O , ~ . U = U, ~x = I ~ l (~x) @ U. Thus we have V ~ I l ~ / , which shows that V e ~ V ~ o e) Let V 6 1 ~ , also ½U e ~ ; and choose U as before.

So we have, with ~(h) = [(hl,h2) = b(hl,a2) + b(al,h2) and r(h) = r(hl,h2) = b(hl,h2)s b(a+h) = b(a) + ~(h) + r ( h ) . is obviously linear, since b is bilinear, and also continuous, since b is continuous. Thus {C~(ElXE2;E3). And r~ R(ElXE2;E 3) by the preceding lemma. This completes the proof. 3. 1) The special case f: IR • E. Proposition. 2) f'(~) Further one = lim then has: f(~ +~)-f(~) . - Proof. 45 We have f((+~) = f(~) + ~(X) + r(~), where ~¢L(IR;E) and r GR(IR;E). l = a and define q: IR---~ E by q(~) ~f(~÷~) - f(~) if~,O, ~ ~f ~ = 0.

E. x ( U and - x ( U o is convex , then z = ~x, where Since ~ O , ~ . U = U, ~x = I ~ l (~x) @ U. Thus we have V ~ I l ~ / , which shows that V e ~ V ~ o e) Let V 6 1 ~ , also ½U e ~ ; and choose U as before. By (c) we have and since ½U is also convex it follows that ½U ~ . 2)). 7) are necessary and sufficient in order that ~ i s the neighborhood-filter of zero for a unique compatible topology on ~ (cf. [Q] ). 9) Proposition. 24 - For any pseudo-topological vector space E, the space E° defined above is a locally convex topological vector space.

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

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