By Marvin J. Greenberg

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For every function and every pair of integers n ~ 0, m ~ 1, put (2) Pn,mU) = sup If(n)(t)1 -m:::;;t~m with flO) = f. Obviously the Pn m are semi-norms on «j "'(R). In order that the functions fa tend to 0 (following a filter 'ff on the set of indices) in «j "'(R) for the topology :Y defined by the semi-norms Pn,m' it is necessary and sufficient that for all integers n ~ 0, the functions fa(n) tend to 0 (following ff) uniformly on every compact subset of R. We say that :Y is the topology of compact convergence for the functions f E «j "'(R) and all their derivatives (cf III, p.

Ixl if 2) Let E and F be two complete, metrisable'Vector spaces over a non-discrete valued division ring, and let 50 be the topology of F. Let 5 be a Hausdorff topology on F, coarser than 50' Show that if the linear mapping u of E in F is continuous for the topology 5 on F, it is still continuous for the topology ·'To on F (use the cor. , p. 19). Deduce that if 5\ and 5 z are two distinct topologies on a vector space E over a non-discrete valued division ring, compatible with the vector space structure of E, and for each of them E is metrisable and complete, then there does not exist a Hausdorff topology on E coarser than 5\ and 52' Give an example of two such topologies on an infinite dimensional vector space E (note that there exist bijections of E on itself such that both the bijection and its inverse are not continuous for a normed space topology on E).

Iii) The relation y - x E P, }vith P a pointed convex cone, is an order relation on E ifand only if P is a proper cone. (i) Axioms (E0 1) and (EOn) imply P + PcP and AP c P for all A > O. As o E P, it follows that P is a pointed convex cone (II, p. II, prop. 10). (ii) Conversely, if P is a pointed convex cone, the relation P + PcP implies that the relation y - x E P is a preorder compatible with the additive group structure ofE (A, VI, p. 3, prop. 3); clearly writing it x y, the set P is identical with the set of x ~ 0; further the relation AP c P for all A ?