Functional Analysis 26 | Open Mapping Theorem [dark version]

The open mapping theorem turns a structural property of linear operators into a topological guarantee: a bounded linear map between Banach spaces sends open sets to open sets exactly when it is surjective. That equivalence matters because “openness” is a strong, usable form of regularity—once it holds, it forces solutions to behave stably under perturbations and underlies many existence-and-uniqueness arguments in analysis.

An open map is defined for metric spaces X and Y: a function F: X → Y is open if for every open set A ⊆ X, the image F(A) is open in Y (with openness understood relative to the metrics on each space). This definition mirrors the logic of continuity—open sets on the input side correspond to open sets on the output side—but it runs in the opposite direction from continuity’s usual “preimage of open is open” condition.

A key example comes from bijections with continuous inverses. If F: X → Y is bijective and its inverse F^{-1}: Y → X is continuous, then F is open. The reason is direct from the definitions: continuity of F^{-1} ensures that whenever B ⊆ Y is open, its preimage (F^{-1})^{-1}(B) = F(B) is open in Y. This example highlights the practical intuition: when a map can be undone continuously, it also preserves the “openness” of sets.

Concrete functions on the real line illustrate what can go right or wrong. The map f(x) = x^3 from R to R is an open map because it is bijective and its inverse is continuous. By contrast, f(x) = x^2 is not an open map. Although it is continuous, it fails surjectivity and injectivity: applying it to the open interval (−2, 2) yields the image (0, 4), which is not open in R because it includes no negative values and does not extend beyond 0. The example shows that continuity alone does not guarantee openness.

With the notion of open maps in place, the open mapping theorem (also called the Banach–Schauder theorem) gives the central result. For a bounded linear operator T between Banach spaces, T is surjective if and only if T is an open map. Completeness of the spaces is essential: the theorem is not just about linearity and boundedness, but about the Banach-space setting where the topology and the operator theory align. The “easy” direction is that surjectivity implies openness; the nontrivial work lies in proving that if T is open, then it must be surjective. The theorem’s power is that it upgrades a linear bounded operator’s surjectivity into a strong topological statement about how it transforms neighborhoods.

A major downstream consequence is the bounded inverse theorem, foreshadowed as the next topic—an application that becomes routine in many areas of functional analysis and differential equations, where one needs not only existence of inverses but control over their boundedness and continuity.