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Bilinear forms and nuclearity. (English) Zbl 0854.46002

For a Banach space \(X\), let \(X_0\) be the space \(X\) endowed with the topology of uniform convergence on compact subsets of \(X^*\), the finest Schwartz topology on \(X\) consistent with the duality \(\langle X,X^* \rangle\). The authors show:
If \(P\) is a Pisier space, then every continuous bilinear form on \(P_0 \times P_0\) is nuclear, moreover \[ P_0 \otimes_\varepsilon P_0= P_0 \otimes_\pi P_0. \] There exists a non-nuclear Fréchet-Schwartz space \(F\) such that
(a) \(F\otimes_\varepsilon F=F \otimes_\pi F\) and
(b) every continuous bilinear form on \(F\times F\) is nuclear.

MSC:

46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
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References:

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