A surjective bounded linear operator between Banach spaces maps open sets to open sets. A consequence of the Baire category theorem, it is one of the foundational results of functional analysis alongside the Hahn-Banach and closed graph theorems.
Facts
StatementIf a bounded, continuous linear operator between two Banach spaces is surjective, then it is an open map, meaning it sends open sets to open sets. 1 Connections
Sources
1. Open Mapping Theorem, Functional Analysis (Wikipedia)
Wikimedia FoundationLead section, opening theorem statementQuote, Lead section, opening theorem statement
In functional analysis, the open mapping theorem, also known as the Banach-Schauder theorem or the Banach theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.
Open Mapping Theorem (MathWorld)
Banach space statementQuote, Banach space statement
A continuous surjective linear mapping between Banach spaces is an open map.
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