Abstract
In [1] Laurent Schwartz introduced the spaces 𝒪M and of multiplication and convolution operators on temperate distributions. Then in [2] Alexandre Grothendieck used tensor products to prove that both 𝒪M and are bornological. Our proof of this property is more constructive and based on duality.
Highlights
The space is dense in each Lq’ q N
Is ultrabornological by Theorems I, 2, and 3. @ The space is ultrabornological as an inductive limit of Frchet spaces
OM C M C the Fourier transformations F: and F: are topological isomorphisms, Theorem 5 follows from Theorems i, 3, and 4
Summary
Washington State University Pullman, Washington 99164-2930, USA In [1] Laurent Schwartz introduced the spaces 0M and of multiplication and convolution operators on temperate distributions. Grothendieck used tensor products to prove that both 0 and are bornological. Our proof of this property is more constructive and based on duality. Multiplication and convolution, inductive and projective limit, bornological, reflexive, and Schwartz spaces. We use C, N, R, and Z, resp., for the set of all complex, nonnegative integer, real, and integer numbers. II’II dual of Lq and by -q the standard norm on L-q the space S of rapidly decreasing functions, resp.
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