Aspects of Mathematics and Its Applications by Jorge Alberto Barroso

By Jorge Alberto Barroso

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Valdivia {85} gave further examples of non-bornological barrelled spaces. 5. Ordered Topological Vector Spaces In his paper [18] Nachbin combines together all three structures we discussed so far and considers ordered topological vector spaces. More precisely, let E be a normed, ordered vector space over IR.. An element x E E is said to be semi-positive if y ~ x implies Ilyll ~ IIxll, and seminegative if y:,,; x implies lIyll~ [r]. Nachbin proves that the space E is isomorphic as a normed, ordered vector space to a subspace of some C€(K), where K is a compact space, if and only if the following two conditions are satisfied: (1) Every element of E is either semi-positive or semi-negative, (2) The set of all elements x ~ 0 is closed.

P. c. Buck call this result 'Nachbin's theorem' in their book ({8, p. 6}) and give a complete proof for it. At the outset of my report on Nachbin's contributions to infinitedimensional holomorphy I must explain the notation introduced by him 46 J. Horvath I Life and Works of L. Nachbin and used universally now. As is well known, if E 1, ••• ,Em and Fare vector spaces over the same field, a map A : (Xl' ••. ,xm)~ A(x l , ... ,xm) from E 1 x ... x Em into F is m-linear if for any index i (1::;; i,,;;;; m) and for arbitrary fixed vectors xj E E j (1::;; j::;; m, j ¥- i) the map Xi ~ A(x l , ••• , Xi' •••• x m ) from E, into F is linear.

Honsith. / Life and Works of L. Nachbin disconnected, and are called stonian if they are also compact. Nachbin became familiar with these spaces through Stone's 1947 lectures in Rio. An element e of an ordered vector space F is said to be an order unit if for every x E F there exists a scalar A;;;. 0 such that - Ae ~ x ~ Ae. If an order unit e exists in F, then the associated norm is defined as above by [z] = inf{AI-Ae ~ x ~ Ae}, With this terminology the theorem of Nachbin and Goodner can be stated as follows: Let F be a Banach space over R.

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