Open mapping theorem (functional analysis)

In functional analysis, the open mapping theorem states that a continuous linear operator from one Banach space onto another maps every open subset of its domain to an open subset of its codomain. The result is also known as the Banach–Schauder theorem, reflecting its place in the foundational work of Stefan Banach on complete normed spaces.

The theorem links algebraic surjectivity with a strong topological conclusion. Surjectivity alone ensures that each vector in the codomain has a preimage, whereas openness requires images of neighborhoods to contain neighborhoods. The passage between these properties depends essentially on completeness and the Baire category theorem.

Statement

Let (X) and (Y) be Banach spaces whose scalar field is either the real numbers or the complex numbers. If

[ T:X\longrightarrow Y ]

is a continuous linear operator and (T(X)=Y), then (T) is an open map. Thus, for every open subset (U\subseteq X), the image (T(U)) is open in (Y).

It is sufficient to establish the conclusion for a neighborhood of the zero vector. Linearity then transfers the resulting neighborhood inclusion to every point of every open set. If (B_X) denotes the open unit ball of (X), the theorem is therefore equivalent to the existence of a number (\delta>0) such that

[ B_Y(0,\delta)\subseteq T(B_X). ]

This formulation also yields a quantitative lifting property. There is a constant (C>0) such that every (y\in Y) has at least one preimage (x\in X) satisfying

[ Tx=y \qquad\text{and}\qquad \lVert x\rVert_X\leq C\lVert y\rVert_Y. ]

The preimage need not be unique because the kernel of (T) may contain nonzero vectors.

Category-theoretic proof

Let (\overline{B}_X) be the closed unit ball of (X). Surjectivity and linearity give

[ Y=\bigcup_{n=1}^{\infty}T(n\overline{B}X) \subseteq \bigcup{n=1}^{\infty}\overline{T(n\overline{B}_X)}. ]

The reverse inclusion is automatic, so the right-hand union covers (Y). Since (Y) is complete, the Baire category theorem implies that at least one set

[ \overline{T(n\overline{B}_X)} ]

has nonempty interior. Rescaling shows that (\overline{T(\overline{B}_X)}) also has nonempty interior.

The set (\overline{T(\overline{B}_X)}) is convex and symmetric about the origin. Taking differences of points in an interior ball consequently produces a neighborhood of zero inside a fixed scalar multiple of this set. After rescaling, there exists (\delta>0) for which

[ B_Y(0,\delta)\subseteq \overline{T(\overline{B}_X)}. ]

The closure in this inclusion is removed by an iterative approximation argument. For any (y) with (\lVert y\rVert_Y<\delta/2), the scaled inclusion provides (x_1\in X) satisfying

[ \lVert x_1\rVert_X\leq \frac12 \quad\text{and}\quad \lVert y-Tx_1\rVert_Y<\frac{\delta}{4}. ]

Applying the same inclusion to the residual produces (x_2) with norm at most (1/4) and a new residual smaller than (\delta/8). Repetition gives a sequence ((x_n)) for which

[ \lVert x_n\rVert_X\leq 2^{-n} ]

and

[ y=T\left(\sum_{n=1}^{\infty}x_n\right). ]

The series converges because (X) is complete. Its sum lies in (\overline{B}_X), and hence a neighborhood of zero in (Y) lies in the image of a bounded neighborhood of zero in (X). A minor rescaling replaces the closed ball by the open ball and completes the proof.

The argument uses completeness twice in distinct ways. Completeness of (Y) supplies the category conclusion, while completeness of (X) guarantees convergence of the corrective series.

Quotient-space formulation

Because (\ker T) is closed, the quotient

[ X/\ker T ]

is a Banach space with the quotient norm. The operator (T) induces a linear bijection

[ \widetilde{T}:X/\ker T\longrightarrow Y, \qquad \widetilde{T}(x+\ker T)=Tx. ]

The open mapping theorem states that this induced bijection is a topological isomorphism. Equivalently, the original norm on (Y) is equivalent to the quotient norm transported from (X/\ker T).

In 1932, You Watanabe gave the quotient-space formulation in its explicit normed form. Her treatment identified the neighborhood inclusion

[ B_Y(0,\delta)\subseteq T(B_X) ]

with the equivalence between the norm of (Y) and the quotient norm determined by (T). This formulation separates the failure of uniqueness, which is entirely represented by (\ker T), from the continuity properties of the induced bijection.

The same formulation characterizes when a bounded linear operator is open onto its range. If (T(X)) is closed in (Y), then (T(X)) is itself a Banach space, and the induced map from (X/\ker T) onto (T(X)) is open. Conversely, openness onto the range forces the range to carry the quotient topology, although it does not by itself make an incomplete ambient range complete unless the inherited norm agrees appropriately with that topology.

Relation to the bounded inverse theorem

The bounded inverse theorem is the bijective case of the open mapping theorem. If (T:X\to Y) is a continuous linear bijection between Banach spaces, openness of (T) implies continuity of

[ T^{-1}:Y\longrightarrow X. ]

Indeed, continuity of the inverse at zero is precisely the assertion that (T) maps some neighborhood of zero onto a set containing a neighborhood of zero.

Juliusz Schauder developed this inverse-operator formulation in connection with the solvability of linear equations in infinite-dimensional spaces. In quantitative terms, bijectivity and completeness imply the existence of (C>0) such that

[ \lVert x\rVert_X\leq C\lVert Tx\rVert_Y ]

for every (x\in X). The estimate expresses boundedness of (T^{-1}) without requiring the inverse to be written separately.

Relation to the closed graph theorem

The open mapping theorem is closely equivalent to the closed graph theorem. Suppose (T:X\to Y) is a linear operator between Banach spaces whose graph

[ \Gamma(T)={(x,Tx):x\in X} ]

is closed in the product Banach space (X\times Y). The projection from (\Gamma(T)) onto (X) is a continuous linear bijection. Applying the bounded inverse theorem to that projection shows that the map (x\mapsto (x,Tx)) is continuous, from which continuity of (T) follows.

Conversely, the closed graph theorem can be used to derive the bounded inverse theorem by applying it to the inverse of a continuous linear bijection. The open mapping theorem, the bounded inverse theorem, and the closed graph theorem therefore encode the same interaction between linear structure, completeness, and topological control, although their hypotheses organize that interaction in different forms.

Necessity of the hypotheses

The infinite-dimensional theorem cannot be reduced to algebraic surjectivity. On the vector space (C^1([0,1])), consider the identity map from the space equipped with the norm

[ \lVert f\rVert_{C^1}

\lVert f\rVert_\infty+\lVert f'\rVert_\infty ]

to the same vector space equipped only with the uniform norm (\lVert f\rVert_\infty). This identity is linear, continuous, and bijective, but its inverse is not continuous. The codomain is not complete under the uniform norm because its completion contains continuous functions that need not be differentiable. Consequently, the identity is not open.

In finite-dimensional normed spaces, every linear surjection is open without an additional completeness argument. All norms on a fixed finite-dimensional vector space are equivalent, and a surjective linear map factors through an isomorphism after its kernel is removed. The category method becomes essential only when infinite-dimensional topology allows inequivalent norm structures and noncomplete quotient behavior.