Vol. 63, No. 1, 1976

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Continuous linear images of pseudo-complete linear topological spaces

Aaron R. Todd

Vol. 63 (1976), No. 1, 281–292

The various properties of topological spaces in the classical Baire category theorems which imply the Baire property also imply the stronger property of pseudo-completeness. In contrast to some of these properties and to the Baire property, J. C. Oxtoby has shown that pseudo-completeness is productive. The following main result places pseudo-completeness in the context of linear topological spaces: Let E and F be linear topological spaces and g be a continuous linear mapping of E into F. If E is pseudo-complete, g is almost open, and the completion of g[E] has a continuous metric, then g[E] is complete. The proof of this result uses the difference theorem, but not an open mapping theorem. The hypotheses lead to a discussion of conditions for a linear mapping to be almost open and for a linear topological space to have a continuous metric.

An example shows that, although a translation invariant continuous metric on a linear topological space E extends to a translation invariant continuous pseudo-metric on the completion of E, this extension need not be a metric, even if it induces a normed topology on E. Other examples show that a pseudo-complete linear topological space need not be complete in its natural uniformity, and that the almost open condition of the main result may not be omitted and is not implied by the combination of the other conditions and the conclusion.

Mathematical Subject Classification 2000
Primary: 46A30
Secondary: 54E99
Received: 7 March 1975
Revised: 4 November 1975
Published: 1 March 1976
Aaron R. Todd