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

A linear map $T:X\to Y$ between two topological vector spaces (TVSs) is called almost open if for any neighborhood $U$ of $0$ in $X,$ the closure of $T(U)$ in $Y$ is a neighborhood of the origin.

Importantly, 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 use this definition.

If $T:X\to Y$ 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 $T:X\to Y$ is almost open then because $T(X)$ is a vector subspace of $Y$ that contains a neighborhood of 0 in $T:X\to Y$ 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 pseudometrizable TVS, $Y$ is a Hausdorff TVS, and $T:X\to Y$ is a closed and almost open linear surjection, then $T$ is an open map.

Theorem: If $T:X\to Y$ is a surjective linear operator from a locally convex space $X$ onto a barrelled space $Y$ then $T$ is almost open.

Theorem: If $T:X\to Y$ is a surjective linear operator from a TVS $X$ onto a Baire space $Y$ then $T$ is almost open.

Theorem: Suppose $T:X\to Y$ 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 $T:X\to Y$ is a surjective open map and $Y$ is a complete metrizable space.

See also

• Almost open set – Difference of an open set by a meager setPages displaying short descriptions of redirect targets
• Barrelled space – Type of topological vector space
• Bounded inverse theorem – Condition for a linear operator to be openPages displaying short descriptions of redirect targets
• Closed graph – Graph of a map closed in the product spacePages displaying short descriptions of redirect targets
• Closed graph theorem – Theorem relating continuity to graphs
• Open set – Basic subset of a topological space
• Open and closed maps – A function that sends open (resp. closed) subsets to open (resp. closed) subsets
• Open mapping theorem (functional analysis) – Condition for a linear operator to be open (also known as the Banach–Schauder theorem)
• Quasi-open map – Function that maps non-empty open sets to sets that have non-empty interior in its codomain
• Surjection of Fréchet spaces – Characterization of surjectivity
• Webbed space – Space where open mapping and closed graph theorems hold

Bibliography

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