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On the existence of a smallest compact support of a seminorm and a linear mapping

Indagationes Mathematicae (Proceedings)
Publication Date
DOI: 10.1016/1385-7258(84)90023-4
  • Mathematics


Abstract Let S be a vector subspace of the vector space scC( E;F) of all continuous mappings of a completely regular space E to a real or complex Hausdorff locally convex space F. A compact subset K of E is a support of a seminorm p on S if, whenever f flying in L vanishes on some neighborhood of K in E, then p( f)=0. In the special case that p = | φ| or p = | u|, where φ is a linear form on S , or more generally, u is a linear mapping of S into a normed linear space, we say that K is a support of φ, or of u, respectively. Sufficient conditions are given in order that, if p has some compact support, then p has a smallest compact support (Proposition 4); and, S being endowed with a locally convex topology, that every continuous p has a smallest compact support (Corollary 7). Such results apply to the vector subspace S (m)( U;F) of S ( U;F) of all mappings of U to F that are continuously m-differentiable, say in the Hadamard, or Fréchet, or other noteworthy senses, where U is a nonvoid open subset of a real locally convex space E; or even to a more general situation, subsuming known examples with an additional nuclearity condition, such as in [10] and other references in the Bibliography (Example 8).

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