Open mapping theorem

The open mapping theorem states that every surjective bounded linear operator between Banach spaces maps open sets to open sets.

Banach space: A normed vector space complete with respect to the metric induced by its norm. Completeness of both spaces is the hypothesis that enables the theorem’s conclusion.
Bounded inverse theorem: A theorem stating that a bounded bijective linear map between Banach spaces has a bounded inverse. It follows by applying openness to a bijection and identifying its inverse images of open sets.
Stefan Banach: A Polish mathematician who helped establish functional analysis and introduced Banach spaces. The theorem belongs to the foundational theory developed in his work.
Fredholm operator: A bounded linear operator with finite-dimensional kernel and cokernel and closed range. Closed-range and solvability arguments for these operators use open mapping principles.
Surjective function: A function whose image equals its entire codomain. Surjectivity ensures the target is covered by images of scaled balls in the domain.
Closed graph theorem: A theorem stating that a linear map between Banach spaces is bounded when its graph is closed. Its standard proof derives boundedness from the open mapping theorem applied to a projection.
Juliusz Schauder: A Polish mathematician known for contributions to functional analysis and topology. His name is associated with the theorem alongside Banach’s.
Linear equation: An equation in which the unknown appears through a linear map. Surjective operators guarantee solutions, while openness controls how target perturbations can be lifted.
Bounded linear operator: A linear map between normed spaces whose output norm is bounded by a constant times the input norm. Continuity and linear scaling let the proof transfer neighborhood estimates across the map.
Quotient space: A vector space formed by treating elements differing by a subspace as equivalent. The theorem makes quotient maps by closed subspaces open, a key structural property.

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