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Aspects of Mathematics and Its Applications by Jorge Alberto Barroso

By Jorge Alberto Barroso

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The last result was given by S. MacLane without proof. In 1959/60 Nachbin lectured on the Haar integral first in Rio de Janeiro and then in Recife. The text of the course was published the same year in Portuguese [32] and English translation appeared in 1965 [51] and was reprinted in 1976. Nachbin gives an unhurried presentation, with many examples, of the theory of Haar measure. The first chapter is an exposition of the H. Cartan-Weil-Bourbaki treatment of integration with respect to a positive Radon measure on a locally compact topological space.

Kakutani proved that such a space F is norm-isomorphic to a space ee(K) for some compact K. The fact that F is Dedekind-complete imposes an additional condition on K: the closure of any open set is open. Spaces which satisfy this last condition are said to be extremally -e Fig. 1. 22 J. 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;;;.

Where a runs through the m! e. satisfies so s = s. faCE,F) such that P(X) = Ax", In this definition m~1 is an integer and Ax" stands for A(x, ... ,x) with X repeated m times. The preceding relation between P and A is expressed by writing P = A o. One has Ax" = s(A)x m , so there is always a symmetric m-linear map A for which P = A o. fase E, F) onto the vector space [j'Ja (mE, F) of all m -homogeneous polynomials, as the polarization formula shows ({26, p. 4}). O-homogeneous polynomials are the constant maps from E into F, their space [j'Ja (0 E, F) can be identified with F The direct sum [j'Ja(E, F) = L m E N [j'Ja(mE, F) is the space of all polynomials from E into F, and its elements can be written uniquely in the form Po + PI + ...

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