Open mapping and closed graph theorems

A bounded linear surjection between Banach spaces maps open sets to open sets. By translation and scaling, this is equivalent to the image of a unit ball containing some ball around zero. The proof starts with Baire applied to countably many dilated images, initially obtaining information about an image closure. A successive-correction argument then constructs an actual preimage as a convergent series, removing the closure. That last step is where completeness of the domain prevents an approximate preimage from disappearing outside the space.

If the surjection is also injective, its inverse is therefore bounded: a bounded linear bijection between Banach spaces is a topological isomorphism. The theorem guarantees a finite inverse bound, not a useful numerical value for it. For a noninjective surjection, the induced map from the Banach quotient X/ker T to Y is a bounded bijection. Consequently distance to the kernel is bounded by a constant times the output norm, even though a specific minimum-norm representative need not exist in a general Banach space.

For a linear map T defined on all of Banach X and taking values in Banach Y, its graph is the set of pairs (x,Tx). If the graph is closed in the product norm topology, it is a Banach subspace of X×Y. Projection from the graph to X is a bounded linear bijection, so the bounded inverse theorem controls its inverse x↦(x,Tx). This proves T bounded. Conversely, a bounded operator has closed graph by taking limits in both coordinates.

The sequential test for graph closedness says that xₙ→x and Txₙ→y imply Tx=y. It does not say that Txₙ converges whenever xₙ converges; demanding that would already assert continuity. The distinction is vital for unbounded operators. A derivative defined on continuously differentiable functions, viewed as a proper subspace of continuous functions with the supremum norm, can have a closed graph in the ambient product yet be unbounded relative to that domain norm. Its domain is not the whole ambient Banach space, so closed graph does not supply a contradictory conclusion.

For a closed operator with proper domain D(T), the graph norm ∥x∥+∥Tx∥ makes its domain Banach when the ambient source and target are Banach. The operator is then bounded from this stronger domain norm into Y. This changes the normed space in the theorem rather than proving the original unbounded operator continuous in the ambient source norm. In differential equations, graph norms encode derivatives that a weaker function norm does not control, making the domain and its regularity part of the analytic data.

Closed range is another separate property. A bounded injective operator can have dense nonclosed range, with an inverse defined on that range that is unbounded in the inherited target norm. If the range were closed, it would be Banach and bounded inverse would apply. Compact diagonal operators on ℓ² offer explicit examples of this distinction. Their small diagonal entries compress coordinate directions and explain inverse instability without violating injectivity, showing why algebraic uniqueness of a solution does not imply a stable solution map.