Almost open linear map

In functional analysis and related areas of mathematics, an almost open linear map between topological vector spacess (TVSs) is a linear operator that satisfies a condition similar to, but weaker than, the condition of being an open map.

Definition

Let T : XY be a linear operator between two TVSs. We say that T is almost open if for any neighborhood U of 0 in X, the closure of T(U) in Y is a neighborhood of the origin.

Note that some authors call T is almost open if for any neighborhood U of 0 in X, the closure of T(U) in T(X) (rather than in Y) is a neighborhood of the origin; this article will not consider this definition.[1]

If T : XY is a bijective linear operator, then T is almost open if and only if T−1 is almost continuous.[1]

Properties

Note that if a linear operator T : XY is almost open then because T(X) is a vector subspace of Y that contains a neighborhood of 0 in Y, T : XY is necessarily surjective. For this reason many authors require surjectivity as part of the definition of "almost open".

Open mapping theorems

Theorem:[1] If X is a complete pseudometrizable TVS, Y is a Hausdorff TVS, and T : XY is a closed and almost open linear surjection, then T is an open map.
Theorem:[1] If T : XY is a surjective linear operator from a locally convex space X onto a barrelled space Y then T is almost open.
Theorem:[1] If T : XY is a surjective linear operator from a TVS X onto a Baire space Y then T is almost open.
Theorem:[1] Suppose T : XY is a continuous linear operator from a complete pseudometrizable TVS X into a Hausdorff TVS Y. If the image of T is non-meager in Y then T : XY is a surjective open map and Y is a complete metrizable space.

See also

References

Bibliography

  • Bourbaki, Nicolas (1950). "Sur certains espaces vectoriels topologiques". Annales de l'Institut Fourier (in French). 2: 5–16 (1951). doi:10.5802/aif.16. MR 0042609.
  • Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7. OCLC 4493665.
  • Jarhow, Hans (1981). Locally convex spaces. Teubner. ISBN 978-3-322-90561-1.CS1 maint: ref=harv (link)
  • Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
  • Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
  • Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
  • Robertson, Alex P.; Robertson, Wendy J. (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. pp. 65–75.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
  • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
  • Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.