An elementary counterexample to the open mapping principle for bilinear maps

Charles Horowitz

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Abstract

In 1.2], Rudin asked whether a continuous bilinear map from the product of two Banach spaces onto a Banach space must be open at the origin; i.e., whether under such a map the image of every neighborhood of zero must contain a neighborhood of zero. Recently, Cohen ll] showed that the answer to the general question was in the negative. However, his counterexample was somewhat involved and left the issue unresolved for bilinear maps on Hilbert spaces. The purpose of this note is to show that the open mapping principle for bilinear maps, as described above, fails even in the finite dimensional case.

Original languageEnglish
Pages (from-to)293-294
Number of pages2
JournalProceedings of the American Mathematical Society
Volume53
Issue number2
DOIs
StatePublished - Dec 1975
Externally publishedYes

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