Almost open linear map

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]

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".

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.

• Barrelled space – A topological vector space with near minimum requirements for the Banach–Steinhaus theorem to hold.
• Bounded inverse theorem
• Closed graph – a graph of a function that is also a closed subset of the product space
• Closed graph theorem
• Open and closed maps – A function that sends open (resp. closed) subsets to open (resp. closed) subsets
• Open mapping theorem (functional analysis) – Theorem giving conditions for a continuous linear map to be an open map (also known as the Banach–Schauder theorem)
• Quasi-open map – A function that maps non-empty open sets to sets that have non-empty interior in its codomain.
• Surjection of Fréchet spaces – A theorem characterizing when a continuous linear map between Fréchet spaces is surjective.
• Webbed space – Topological vector spaces for which the open mapping and closed graphs theorems hold

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