By Jorge Alberto Barroso
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Z" is said to be of finite V'-type if there exist numbers M > 0 and T > 0 such that If(z)1 , ; ; MV'( Tlz Dfor z E C. The greatest lower bound of all numbers T for which this inequality holds is called the V'-type of f. Nachbin proves that the V'-type of f is given by limsup,... 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.
X) and C€(vX), by J. Schmets, M. de Wilde and others. (X, E) of continuous functions with values in a topological vector space E, where not only X but also E determines the properties of C€(X, E), and has written two monographs {71}, {72} which give an account of the work of several authors. Buchwalter also has a set of informative lecture notes on the subject {14}. On the other hand Y. Kornura and later M. 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.
For a later purpose I will describe a more general form of this theorem given by E. Bishop at about the same time when Nachbin started 1 27 J. Horvath I Life and Works of L. Nachbin publishing on weighted approximation (1%1). Bishop considers a subalgebra d of cg(K) and calls a set M c K antisymmetric with respect to d if I(x) == I(y) for all x, y E M and for all real-valued functions in d. If d is self-adjoint, in particular if the functions in d are real-valued, then M is antisymmetric if and only if all functions in d are constant on M.