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 : X → Y 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 : X → Y 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 : X → Y is almost open then because T(X) is a vector subspace of Y that contains a neighborhood of 0 in Y, T : X → Y 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 : X → Y is a closed and almost open linear surjection, then T is an open map.
- Theorem:[1] If T : X → Y 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 : X → Y is a surjective linear operator from a TVS X onto a Baire space Y then T is almost open.
- Theorem:[1] Suppose T : X → Y is a continuous linear operator from a complete pseudometrizable TVS X into a Hausdorff TVS Y. If the image of T is non-meager then Suppose T : X → Y is a surjective open map and Y is a complete pseudo-metrizable space.
See also
- Barrelled space
- Open mapping theorem (functional analysis) (also known as the Banach–Schauder theorem)
References
- "Sur certains espaces vectoriels topologiques" (in French). Annales de l'Institut Fourier 2: 5–16 (1951). 1950. http://www.numdam.org/item?id=AIF_1950__2__5_0.
- 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.
- Khaleelulla, S. M. (July 1, 1982). written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
- 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. (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 (August 6, 2006). Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
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