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.

If T : X → Y is a bijective linear operator, then T is almost open if and only if T^-1 is almost continuous.

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: If X is a complete pseudo-metrizable 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: 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: If T : X → Y is a surjective linear operator from a TVS X onto a Baire space Y then T is almost open.

Theorem: Suppose T : X → Y is a continuous linear operator from a complete pseudo-metrizable 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

• Bourbaki, Nicolas (1950). "Sur certains espaces vectoriels topologiques". Annales de l'Institut Fourier (in French). 2: 5–16 (1951). MR 0042609.
• Robertson, Alex P.; Robertson, Wendy J. (1964). Topological vector spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge University Press. pp. 65–75.
• Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7. OCLC 4493665. {{cite book}}: Invalid |ref=harv (help)
• Jarhow, Hans (1981). Locally convex spaces. Teubner. ISBN 978-3-322-90561-1. {{cite book}}: Invalid |ref=harv (help)
• Khaleelulla, S. M. (1982). Written at Berlin Heidelberg. Counterexamples in topological vector spaces. GTM. Vol. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370. {{cite book}}: Invalid |ref=harv (help)
• Narici, Lawrence (2011). Topological vector spaces. Boca Raton, FL: CRC Press. ISBN 1-58488-866-0. OCLC 144216834.
• Schaefer, Helmut H. (1971). Topological vector spaces. GTM. Vol. 3. New York: Springer-Verlag. p. 60. ISBN 0-387-98726-6.
• Schaefer, Helmut H. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. {{cite book}}: Invalid |ref=harv (help)
• Treves, Francois (2006). Topological vector spaces, distributions and kernels. Mineola, N.Y: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. {{cite book}}: Invalid |ref=harv (help)