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 in Y then T : X → Y is a surjective open map and Y is a complete metrizable space.

See also

References
  1. Narici 2011, pp. 466-468.
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  • Trèves, François (August 6, 2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
  • Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.CS1 maint: ref=harv (link)
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