# The Converse of Minlos' Theorem : Dedicated to Professor Tsuyoshi Ando on the occasion of his sixtieth birthday

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## Abstract

Let \mathscr{M} be the class of barrelled locally convex Hausdorff space <I>E</I> such that <I>E</I><SUB><I>b</I></SUB>' satisfies the property <I>B</I> in the sense of Pietsch. It is shown that if <I>E</I>∈\mathscr{M} and if each continuous cylinder set measure on <I>E'</I> is σ(<I>E'</I>, <I>E</I>) -Radon, then <I>E</I> is nuclear. There exists an example of non-nuclear Fréchet space <I>E</I> such that each continuous Gaussian cylinder set measure on is <I>E'</I> is σ(<I>E'</I>, <I>E</I>)-Radon. Let <I>q</I> be 2≤ <I>q</I><∞. Suppose that <I>E</I>∈\mathscr{M} and <I>E</I> is a projective limit of Banach space {<I>E</I><SUB>α</SUB>} such that the dual <I>E</I><SUB>α</SUB>' is of cotype <I>q</I> for every α. Suppose also that each continuous Gaussian cylinder set measure on <I>E'</I> is σ(<I>E'</I>, <I>E</I>)-Radon. Then <I>E</I> is nuclear.

## Journal

Publications of the Research Institute for Mathematical Sciences   [List of Volumes]