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Radonifying Mappings: a Counterexample

Radonifying Mappings: a Counterexample D. J. H. GARLING We shall prove the following: THEOREM. There exist reflexive Banach spaces E and F, and a continuous linear mapping T from E into F which is \-Radonifying but not O-Radonifying. This provides a negative answer to a problem raised by Schwartz [3; p. XII.6]. We shall use the terminology and notation of [3]. Let S be a continuous linear mapping from a Banach space G into a Banach space H which is 1-Radonifying but not approximately O-Radonifying—for example, 00 1 take G = Z , H = Z and let S be the diagonal mapping defined by 5(0^) ) = (<r,- x ), where a = n~^(\ogn)~ . Then £* ff < oo so that S is nuclear, and so certainly n = 1 n 1-Radonifying. On the other hand, £" ff(l — log<r ) = oo, so that S is not = 1 n n O-Radonifying [2; Thiorem e 3], and therefore not approximately O-Radonifying [3; Proposition (XVII, 3; 1)]. This means [3; Thioreme (XVI, 2; 1)] that there exists /? > 0, such that for each n there exists a Radon probability measure l supported by finitely many points of G such that http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Radonifying Mappings: a Counterexample

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/3.2.167
Publisher site
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Abstract

D. J. H. GARLING We shall prove the following: THEOREM. There exist reflexive Banach spaces E and F, and a continuous linear mapping T from E into F which is \-Radonifying but not O-Radonifying. This provides a negative answer to a problem raised by Schwartz [3; p. XII.6]. We shall use the terminology and notation of [3]. Let S be a continuous linear mapping from a Banach space G into a Banach space H which is 1-Radonifying but not approximately O-Radonifying—for example, 00 1 take G = Z , H = Z and let S be the diagonal mapping defined by 5(0^) ) = (<r,- x ), where a = n~^(\ogn)~ . Then £* ff < oo so that S is nuclear, and so certainly n = 1 n 1-Radonifying. On the other hand, £" ff(l — log<r ) = oo, so that S is not = 1 n n O-Radonifying [2; Thiorem e 3], and therefore not approximately O-Radonifying [3; Proposition (XVII, 3; 1)]. This means [3; Thioreme (XVI, 2; 1)] that there exists /? > 0, such that for each n there exists a Radon probability measure l supported by finitely many points of G such that

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Bulletin of the London Mathematical Society – Wiley

Published: Jul 1, 1971

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