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Open mapping theorem (functional analysis)

The open mapping theorem, also known as the Banach–Schauder theorem, is a foundational result in functional analysis asserting that every surjective continuous (or bounded) linear operator between two Banach spaces maps open sets to open sets.^[1] This property ensures that the operator preserves the topological structure in a strong sense, distinguishing Banach spaces from more general normed spaces where counterexamples exist.^[1] First proved by Juliusz Schauder in 1930 using the Baire category theorem, the result built on earlier work by Stefan Banach on linear operations and inverse mappings in normed spaces.^[2] The proof typically demonstrates that the image of the unit open ball in the domain contains a non-empty open ball in the codomain, leveraging completeness to apply the Baire category theorem to a covering of the unit ball by preimages of scaled balls.^[1] This core argument extends by linearity to show openness for arbitrary open sets, highlighting the role of completeness in Banach spaces.^[1] Among the cornerstone theorems of functional analysis—alongside the Hahn–Banach theorem and the uniform boundedness principle—the open mapping theorem has profound implications for operator theory.^[3] It implies the inverse mapping theorem, stating that the inverse of a bijective continuous linear operator between Banach spaces is automatically continuous, and the closed graph theorem, which equates boundedness with the closedness of the operator's graph.^[1] These corollaries underpin applications in partial differential equations, approximation theory, and the study of dual spaces, ensuring structural stability in infinite-dimensional settings.^[1]

Background Concepts

Banach spaces

A Banach space is a complete normed vector space over the real or complex numbers. A normed vector space consists of a vector space X equipped with a norm \|\cdot\|, which is a function from X to [0, \infty) satisfying the following axioms for all x, y \in X and scalars \alpha:

\|x\| = 0 if and only if x = 0,
\|\alpha x\| = |\alpha| \|x\|,
\|x + y\| \leq \|x\| + \|y\| (triangle inequality).^[4] The norm induces a metric d(x, y) = \|x - y\|, turning X into a metric space.^[5]

Completeness in a Banach space means that every Cauchy sequence in X converges to an element in X, where a sequence \{x_n\} is Cauchy if for every \epsilon > 0, there exists N such that \|x_m - x_n\| < \epsilon for all m, n > N. This completeness is with respect to the metric derived from the norm, and thus equivalent to the space being complete as a normed space, since Cauchy sequences in the norm coincide with those in the metric.^[5] Normed spaces without completeness, such as the space of rational sequences under the \ell^2 norm, serve as precursors but lack this convergence property.^[5] Key properties of Banach spaces include the fact that closed subsets, such as the closed unit ball B = \{x \in X : \|x\| \leq 1\}, are themselves complete, inheriting completeness from the ambient space.^[5] This ensures that limits of convergent sequences remain within the space, which is essential for handling approximations and infinite-dimensional phenomena where finite-dimensional intuitions like compactness of the unit ball fail.^[5] Prominent examples of Banach spaces include finite-dimensional Euclidean spaces \mathbb{R}^n or \mathbb{C}^n under the Euclidean norm, which are complete due to their finite dimensionality.^[6] Infinite-dimensional instances encompass the sequence spaces \ell^p for $1 \leq p \leq \infty, consisting of sequences (x_n) with \|x\|_p = \left( \sum |x_n|^p \right)^{1/p} < \infty (or sup norm for p = \infty), which are complete under this norm. Similarly, the function spaces L^p(\mu) over a measure space (\Omega, \mu), comprising equivalence classes of measurable functions f with \|f\|_p = \left( \int |f|^p \, d\mu \right)^{1/p} < \infty, form Banach spaces for $1 \leq p \leq \infty.^[7] Completeness is crucial in infinite-dimensional Banach spaces because it guarantees the convergence of Cauchy sequences arising from limits of approximations, such as partial sums or integrals, preventing pathologies like non-convergent bounded sequences that occur in incomplete settings.^[5] This property underpins the reliability of analytic tools in unbounded domains.^[5]

Bounded linear operators

A linear operator T: X \to Y between normed linear spaces X and Y is a function satisfying T(\alpha x + \beta z) = \alpha T x + \beta T z for all scalars \alpha, \beta and vectors x, z \in X. Such an operator is bounded if there exists a constant C < \infty such that \|T x\|_Y \leq C \|x\|_X for all x \in X, or equivalently, if \sup_{\|x\|_X \leq 1} \|T x\|_Y < \infty.^[8]^[9] This boundedness condition is equivalent to continuity of T at the origin $0 \in X, and hence continuity everywhere, since linearity implies that continuity at one point extends to the entire space.^[8]^[9] The operator norm of a bounded linear operator T is defined as \|T\| = \sup_{\|x\|_X \leq 1} \|T x\|_Y, which is the smallest constant C satisfying the boundedness inequality and is itself a norm on the space of bounded operators.^[8]^[9] This norm satisfies submultiplicativity: for bounded operators T: X \to Y and S: Y \to Z, \|S \circ T\| \leq \|S\| \|T\|.^[9] Examples of bounded linear operators include the identity operator I: X \to X, which satisfies \|I x\| = \|x\| and thus has operator norm \|I\| = 1.^[5] Another is the Fredholm integral operator on the Banach space C([0,1]), defined by (K f)(x) = \int_0^1 k(x,y) f(y) \, dy for continuous kernel k, which is bounded with \|K\| = \max_{0 \leq x \leq 1} \int_0^1 |k(x,y)| \, dy.^[8]^[5] In contrast, unbounded linear operators do not satisfy the boundedness condition and thus have infinite operator norm; they are typically defined only on dense subspaces of the domain to ensure well-definedness, such as the differentiation operator on smooth functions in C^\infty([0,1]), where eigenfunctions like e^{\lambda x} yield arbitrarily large ratios \|u'\| / \|u\| = |\lambda|.^[5] While the open mapping theorem concerns surjective bounded operators between Banach spaces, the study of bounded operators more generally applies to normed spaces.^[9]

Statement of the Theorem

Classical formulation

The open mapping theorem, in its classical formulation, asserts that if X and Y are Banach spaces over the same field and T: X \to Y is a surjective bounded linear operator, then T is an open mapping.^[10] This means that for every open set U in X, the image T(U) is an open set in Y.^[10] The theorem establishes a fundamental property of continuous linear operators between complete normed spaces, ensuring that surjectivity implies openness in the topological sense.^[10] An equivalent reformulation of the theorem highlights the behavior on the open unit ball: T maps the open unit ball B_X = \{x \in X : \|x\| < 1\} to a set in Y that contains some open ball centered at the origin.^[10] Specifically, there exists \varepsilon > 0 such that the open ball B_Y(0, \varepsilon) = \{y \in Y : \|y\| < \varepsilon\} is contained in T(B_X).^[10] This local surjectivity around zero extends to the global openness of T due to the homogeneity and additivity of linear operators.^[10] The theorem relies on the completeness of both spaces, distinguishing it from mappings in more general topological vector spaces.^[10]

Transpose and quantitative formulations

A quantitative version of the open mapping theorem provides an explicit measure of openness for surjective bounded linear operators between Banach spaces. Specifically, if T: X \to Y is a surjective bounded linear operator, where X and Y are Banach spaces, then there exists a constant \delta > 0 such that the open ball B_Y(0, \delta) is contained in the image under T of the open unit ball B_X(0, 1).^[11] This inclusion quantifies how much T "expands" neighborhoods near the origin, ensuring uniform control over the preimages. Equivalently, for surjective T, the quantity \inf_{\|y\|_Y = 1} \inf \{ \|x\|_X : T x = y \} > 0, which bounds the norm of solutions from below relative to the data.^[11] The transpose formulation offers a dual characterization using the adjoint operator. For a bounded linear operator T: X \to Y between Banach spaces X and Y, T is open (equivalently, surjective) if and only if its adjoint T^*: Y^* \to X^* is injective, where X^* and Y^* denote the continuous dual spaces.^[12] This follows from the fact that surjectivity of T implies T^* is bounded below and hence injective. These variants relate to the concept of uniform openness, where the quantitative estimate directly implies the classical openness by scaling arguments: if a neighborhood of the origin maps to contain a smaller neighborhood, then arbitrary open sets map onto open sets. Conversely, openness yields the quantitative form via compactness of the unit sphere or Baire category considerations, though the details lie beyond the scope here. The quantitative aspects gained emphasis in later pedagogical treatments for clarifying estimates in applications.^[11]

Proof of the Theorem

Preliminary lemmas

The proof of the open mapping theorem relies on several supporting lemmas that establish key properties of bounded linear operators between Banach and normed spaces. These lemmas focus on surjectivity criteria, the openness of nearly open maps, and the structure of preimages under surjective operators, setting the stage for the application of the Baire category theorem. A foundational construction concerns the absorbing sets formed by preimages of balls under a surjective bounded linear operator T: X \to Y, where X is a Banach space and Y is a normed space. Since T is surjective, every x \in X satisfies \|Tx\|_Y < n for sufficiently large n \in \mathbb{N}, so X = \bigcup_{n=1}^\infty T^{-1}(B_Y(0, n)), where B_Y(0, n) = \{ y \in Y : \|y\|_Y < n \}. Each set T^{-1}(B_Y(0, n)) is open in X, and by linearity of T, T^{-1}(B_Y(0, n)) = n \cdot T^{-1}(B_Y(0, 1)). This scaled covering provides the basis for completeness-based arguments in the main proof.^[13] Schauder's lemma gives a sufficient condition for surjectivity in terms of the image of the unit ball. Schauder's lemma. Let T: X \to Y be a bounded linear operator from a Banach space X to a normed space Y. If T(B_X(0, 1)) contains a neighborhood of the origin in Y, then T is surjective.^[13] To see this, suppose B_Y(0, \epsilon) \subset T(B_X(0, 1)) for some \epsilon > 0. For arbitrary y \in Y, choose r > \|y\|_Y / \epsilon. Then T(B_X(0, r)) = r \, T(B_X(0, 1)) \supset r B_Y(0, \epsilon) = B_Y(0, r \epsilon), which contains y. Thus, y \in T(X). Another key result addresses nearly open maps, bridging the gap between closures of images and actual openness. Lemma on nearly open maps. Let T: X \to Y be a bounded linear operator from a complete normed space X to a normed space Y. If for every r > 0, there exists \delta > 0 such that B_Y(0, \delta) \subset \overline{T(B_X(0, r))} (i.e., T is nearly open), then T is open.^[14] This lemma ensures that if the closures of scaled images absorb balls around the origin, the operator maps open sets to open sets, leveraging completeness of X. The proof involves constructing points in the image via limits and linearity, but details are deferred to the main argument.

Baire category theorem application

The proof of the open mapping theorem proceeds by applying the Baire category theorem to establish that a surjective bounded linear operator T: X \to Y between Banach spaces X and Y has the property that the closure of the image of the open unit ball in X contains an open ball in Y around the origin, and then invoking the lemma on nearly open maps to conclude openness.^[15] Since T is linear and continuous, it suffices to show that there exists \epsilon > 0 such that the open unit ball B_Y(0, \epsilon) in Y is contained in T(B_X(0, 1)), where B_X(0, 1) = \{ x \in X : \|x\|_X < 1 \}.^[1] Because T is surjective, every element of Y lies in the image of some scaled multiple of the open unit ball in X. Specifically, Y = \bigcup_{n=1}^\infty T(n B_X(0, 1)), where B_X(0, 1) is the open unit ball. The sets \overline{T(n B_X(0, 1))} are closed in Y and form a covering of Y.^[15] The space Y, being a Banach space, is a complete metric space and thus a Baire space, meaning it cannot be expressed as a countable union of nowhere dense sets.^[15] Therefore, at least one of these closed sets, say for n = J, must have nonempty interior: there exist c \in Y and r > 0 such that the open ball B_Y(c, r) \subset \overline{T(J B_X(0, 1))}.^[1] This containment in the closure implies the nearly open condition from the preliminary lemma. Scaling appropriately and applying the lemma on nearly open maps (whose proof constructs exact preimages via convergent series in the complete space X) yields B_Y(0, \epsilon) \subset T(B_X(0, 1)) for some \epsilon = r/(4J) > 0.^[1] The completeness of Y is essential, as it ensures Y is a Baire space via the Baire category theorem; without completeness, Y could be meager (a countable union of nowhere dense sets), allowing the closed sets \overline{T(n B_X(0, 1))} to cover Y without any having nonempty interior, which would prevent the derivation of the open ball containment and lead to failures of openness, as illustrated in counterexamples for incomplete normed spaces.^[15] This application integrates preliminary lemmas on the continuity and surjectivity of T to confirm the covering properties of the sets involved.^[1]

Examples and Counterexamples

Positive examples in Banach spaces

A canonical positive example of the open mapping theorem arises with the identity operator on the separable Hilbert space \ell^2 of square-summable real or complex sequences, equipped with the standard \ell^2-norm \|x\|_2 = \left( \sum_{n=1}^\infty |x_n|^2 \right)^{1/2}. This operator I: \ell^2 \to \ell^2, defined by I(x) = x for all x \in \ell^2, is bounded and linear with operator norm \|I\| = 1, and it is bijective, hence surjective. By the open mapping theorem, I maps open sets to open sets; in particular, the image I(B(0,1)) of the open unit ball B(0,1) = \{ x \in \ell^2 \mid \|x\|_2 < 1 \} is the open unit ball itself, which contains every neighborhood of the origin in \ell^2. The quantitative version of the theorem guarantees the existence of some m > 0 such that m B(0,1) \subset I(B(0,1)); here, m = 1 works directly, as the unit ball coincides with its image and absorbs scaled versions of itself.^[16] The Fourier transform provides another illustrative example on the Banach space L^2(\mathbb{R}) of square-integrable functions on the real line, with the L^2-norm \|f\|_2 = \left( \int_\mathbb{R} |f(x)|^2 \, dx \right)^{1/2}. The Fourier transform \mathcal{F}: L^2(\mathbb{R}) \to L^2(\mathbb{R}), given initially for suitable functions by (\mathcal{F} f)(\xi) = \int_\mathbb{R} f(x) e^{-2\pi i x \xi} \, dx and extended by density and continuity, is a bounded linear operator with \|\mathcal{F}\| = 1. Plancherel's theorem establishes that \mathcal{F} is unitary, meaning it is surjective and preserves the inner product (hence the norm), so its inverse is also bounded. Consequently, the open mapping theorem implies that \mathcal{F} is open: the image of the open unit ball B_{L^2}(0,1) is again the open unit ball, which is open in L^2(\mathbb{R}). Quantitatively, m = 1 holds, as \mathcal{F} isometrically maps the unit ball onto itself, ensuring it contains scaled neighborhoods around zero.^[17]

Failures in incomplete spaces

The completeness of the domain space is a crucial hypothesis in the open mapping theorem, as its absence allows for bounded linear surjective operators that fail to map open sets to open sets. Without completeness, the Baire category theorem cannot be applied in the proof, leading to such failures. A standard counterexample is the diagonal operator on the incomplete normed space c_{00}, the vector space of all sequences x = (x_n)_{n=1}^\infty with only finitely many non-zero terms, equipped with the supremum norm \|x\|_\infty = \sup_n |x_n|. This space is incomplete, as it is dense in the Banach space c_0 of sequences converging to zero but not closed therein.^[18] Consider the linear operator T: c_{00} \to c_{00} defined by (Tx)_n = \frac{x_n}{n} for each n \in \mathbb{N}. This operator is bounded, since \|Tx\|_\infty = \sup_n \left| \frac{x_n}{n} \right| \le \sup_n |x_n| = \|x\|_\infty, so \|T\| = 1. It is also surjective: for any y \in c_{00}, the sequence x = (n y_n)_{n=1}^\infty has finite support and satisfies Tx = y.^[19] However, T is not open. The image of the closed unit ball B = \{ x \in c_{00} : \|x\|_\infty \le 1 \} is the set T(B) = \{ y \in c_{00} : \sup_n n |y_n| \le 1 \}. This set does not contain any open ball around the origin. Indeed, for any r > 0, consider the sequence y^{(k)} with y^{(k)}_k = \frac{r}{2} and y^{(k)}_n = 0 for n \ne k. Then \|y^{(k)}\|_\infty = \frac{r}{2} < r, so y^{(k)} \in B_{c_{00}}(0, r), but k \cdot \frac{r}{2} > 1 for sufficiently large k, so y^{(k)} \notin T(B). Thus, no ball around 0 is contained in T(B), confirming that T fails to be open.^[20] These examples illustrate that incompleteness undermines the Baire category argument in the theorem's proof, allowing the unit ball image to absorb the origin without containing a neighborhood.

Consequences and Applications

Bounded inverse theorem

The bounded inverse theorem states that if X and Y are Banach spaces and T: X \to Y is a bijective bounded linear operator, then the inverse operator T^{-1}: Y \to X is also bounded (and hence continuous).^[21]^[1] To see this, note first that bijectivity implies surjectivity of T. By the open mapping theorem, T maps open sets in X to open sets in Y. Thus, for any open set U \subset X, T(U) is open in Y, which means T^{-1} maps open sets in Y to open sets in X, establishing the continuity of T^{-1}. Since T^{-1} is a continuous linear operator between Banach spaces, it is bounded.^[1] A quantitative version follows from the corresponding estimate in the open mapping theorem. There exists a constant \delta > 0 such that the image under T of the closed unit ball in X contains the open ball of radius \delta in Y. For the inverse, this implies \|T^{-1} y\|_X \leq (1/\delta) \|y\|_Y for all y \in Y, so \|T^{-1}\| \leq 1/\delta.^[22] This theorem ensures well-posedness for linear equations Tx = y in Banach spaces: if T is bijective and bounded, then solutions exist, are unique, and depend continuously on the right-hand side y with respect to the norms.^[23]

Closed graph theorem

The closed graph theorem provides a characterization of bounded linear operators between Banach spaces in terms of the topological properties of their graphs. Specifically, let X and Y be Banach spaces over the same field, and let T: X \to Y be a linear operator defined on the entire space X. Then T is bounded (equivalently, continuous) if and only if its graph G(T) = \{ (x, Tx) \mid x \in X \} is a closed subspace of the product space X \times Y equipped with the product norm \|(x, y)\| = \|x\|_X + \|y\|_Y.^[24] The direction that boundedness implies a closed graph follows readily from continuity: if (x_n, T x_n) \to (x, y) in X \times Y, then x_n \to x and T x_n \to y, so y = T x by continuity of T. The converse, that a closed graph implies boundedness, relies on the open mapping theorem. To see this, note that G(T) is itself a Banach space under the product norm, and the natural projection \pi_1: G(T) \to X given by \pi_1(x, Tx) = x is a bijective continuous linear map between Banach spaces. By the open mapping theorem, \pi_1^{-1} is continuous. Composing with the continuous projection \pi_2: G(T) \to Y onto the second coordinate yields T = \pi_2 \circ \pi_1^{-1}, which is therefore continuous (and hence bounded).^[25]^[24] An illustrative example of an operator whose graph is not closed arises from any unbounded linear operator defined on the whole space, as the closed graph theorem implies that unboundedness precludes closure of the graph. Such operators, including discontinuous linear functionals, exist on any infinite-dimensional Banach space but require the axiom of choice for their construction via Hamel bases; explicit descriptions are impossible in standard models of mathematics. For instance, on \ell^2, a linear functional that is unbounded on the unit ball would have a graph that fails to contain limits of sequences from the graph, such as those approaching a point outside G(T).^[26]^[27] This theorem is closely intertwined with the open mapping theorem, often serving as its corollary by interchanging the roles of the identity and the operator in an auxiliary mapping, thereby reducing the problem of graph closure to openness properties.^[24]

Other implications

The uniform boundedness principle can be derived as an indirect consequence of the open mapping theorem through the closed graph theorem, establishing that pointwise bounded families of continuous linear operators from a Banach space to normed spaces are uniformly bounded.^[28] A key corollary is the closed range theorem, which states that for a bounded linear operator T between Banach spaces X and Y, the range of T is closed in Y if and only if the range of the adjoint operator T^* is closed in X^*.^[28] In the context of short exact sequences of Banach spaces $0 \to A \xrightarrow{i} B \xrightarrow{p} C \to 0 with bounded linear operators, the open mapping theorem implies that surjectivity of p: B \to C ensures p is an open mapping, providing topological exactness.^[29] These implications find applications in solving partial differential equations (PDEs), where the open mapping theorem guarantees the existence and stability of solutions for surjective differential operators on Banach or Hilbert spaces of functions, such as in elliptic boundary value problems.^[30] In operator theory on Hilbert spaces, it aids in analyzing spectral properties and invertibility of unbounded operators, ensuring closed ranges for self-adjoint extensions relevant to quantum mechanics.^[30]

Historical Development

Origins and Banach's contribution

The open mapping theorem emerged during the 1920s and 1930s as part of the foundational developments in functional analysis, a field that was rapidly evolving through investigations into linear operators on infinite-dimensional spaces. David Hilbert's earlier work on integral equations and spectral theory in the early 1900s laid groundwork for operator studies, while John von Neumann's contributions in the late 1920s and early 1930s, particularly on Hilbert spaces and rings of operators in the context of quantum mechanics, emphasized the need for rigorous topological properties of these spaces.^[31]^[32] Stefan Banach played a pivotal role in formalizing the theorem; his work in 1929 on inverse mappings laid foundational groundwork using set-theoretic methods. The open mapping theorem was proved by Juliusz Schauder in 1930 using the Baire category theorem from 1907, building on Banach's ideas. It is named the Banach–Schauder theorem because Schauder provided the first proof using the Baire category theorem for operators between Banach spaces.^[31]^[33]^[2] This proof addressed the openness of images under surjective continuous linear operators between complete normed spaces, marking a cornerstone in the axiomatization of functional analysis. Banach's 1932 monograph Théorie des opérations linéaires presented the theorem systematically alongside other key results, establishing it as a fundamental principle for linear operations in Banach spaces—which are named after him for his 1920 dissertation introducing complete normed linear spaces.^[31]^[34] The theorem built on prior influences, including Baire's category theorem for its topological arguments and Juliusz Schauder's 1930 fixed-point theorem, which advanced compactness in Banach spaces and complemented Banach's completeness requirements.^[33]^[35] Initial proofs were developed in this collaborative Polish school of mathematics in Lwów, and the result is often credited as the Banach–Schauder theorem due to Schauder's contributions to basis theory and operator compactness.^[34]^[36]

Subsequent extensions

Following Banach's foundational work in 1932, subsequent developments in the mid-20th century extended the open mapping theorem to alternative proof strategies and broader contexts. One notable alternative approach leverages Michael's selection theorem, introduced in 1956, which guarantees the existence of continuous selections for lower hemicontinuous set-valued maps from paracompact spaces to Banach spaces with convex, closed values. This theorem has been combined with the open mapping principle to establish surjective selections and strong open mapping results for set-valued operators, particularly in cone structures over Banach spaces.^[37] Quantitative refinements emerged in the 1940s and were further elaborated in later decades, providing explicit bounds absent in the classical qualitative statement. Mahlon M. Day's work on normed linear spaces in the 1940s laid groundwork for such refinements, emphasizing uniform boundedness in solvability estimates for surjective operators between Banach spaces. A modern exposition appears in Terence Tao's 2009 notes, where the theorem is reformulated to equate surjectivity with the existence of a constant C > 0 such that for every f \in Y, there is u \in X with Lu = f and \|u\|_X \leq C \|f\|_Y, highlighting the quantitative solvability with explicit constant dependence derived via the Baire category theorem.^[38] The theorem gained formal recognition in the 1950s through its inclusion in Nicolas Bourbaki's Éléments de mathématique, specifically in the chapters on topological vector spaces published between 1953 and 1955, where it serves as a cornerstone for results on continuous linear operators.^[39] These developments also fostered connections to nonlinear analysis, as Michael's framework extends openness properties to multivalued nonlinear mappings, influencing fixed-point and equilibrium theories in infinite-dimensional settings. As of 2025, no major theoretical updates to the core theorem have appeared, though it underpins modern applications in machine learning, particularly in neural operator architectures that approximate continuous mappings between Banach function spaces for solving partial differential equations.^[40] For instance, universal approximation theorems for neural operators operate within Banach spaces to guarantee effective learning of operator solutions.

Generalizations

To Fréchet spaces

A Fréchet space is a topological vector space that is complete and metrizable, with its topology induced by a countable family of seminorms that separate points.^[41] This structure generalizes Banach spaces, where the topology arises from a single norm, to settings defined by multiple seminorms, allowing for more flexible topologies on spaces like function spaces.^[41] The open mapping theorem extends to Fréchet spaces as follows: if F and E are Fréchet spaces and T: F \to E is a surjective continuous linear operator, then T is an open mapping.^[41] Banach spaces form a special case of this result, where the single norm defines the metric.^[41] The proof adapts the Banach space version by leveraging the Baire category theorem, which applies because Fréchet spaces are complete metric spaces. Specifically, the codomain E can be covered by a countable union of the images under T of scaled closed balls in F, and by the Baire category theorem, one such image has nonempty interior, ensuring that the image of any open ball in F contains an open ball in E.^[41] A representative example is the space C^\infty(\Omega) of smooth functions on an open set \Omega \subseteq \mathbb{R}^d, equipped with seminorms \|u\|_k = \max_{|\alpha| \leq k, x \in K_k} |\partial^\alpha u(x)|, where K_k are compact exhaustion sets of \Omega and \alpha are multi-indices. This space is Fréchet, and the theorem applies to surjective continuous linear operators between such spaces, such as differential operators under suitable conditions.^[42]

To barrelled topological vector spaces

A barrelled topological vector space is defined as a topological vector space in which every barrel—a closed, convex, balanced, and absorbing set—is a neighborhood of the origin.^[43] The open mapping theorem generalizes to barrelled spaces as follows: if X is a barrelled topological vector space and Y is a locally convex topological vector space, then every surjective continuous linear operator T: X \to Y is open.^[43] This result relies on the barrelled topology of the domain to ensure that the image of open sets covers neighborhoods in the codomain appropriately.^[43] In certain formulations, additional conditions such as completeness of X or Y are included to invoke Baire category principles for the proof. Fréchet spaces, being complete metrizable locally convex spaces, constitute a significant subclass of barrelled spaces.^[43] In complete barrelled spaces, the open mapping theorem particularly applies to quotient maps: the canonical surjection from a complete barrelled space onto its quotient by a closed hyperplane or subspace is open, preserving the topological structure under the quotient topology.^[43] The barrelledness of the domain is crucial, as the theorem fails in non-barrelled topological vector spaces, where surjective continuous linear maps need not be open.

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[PDF] arXiv:2310.12683v2 [quant-ph] 3 Jun 2024
Jun 3, 2024 · ... Fourier transform to a unitary map from L2(R) to itself. This definition in L2(R) coincides with the integral definition when f is in L2(R) ∩ L1 ...
[18]
[PDF] Functional Analysis, Sobolev Spaces and Partial Differential Equations
... Adjoint ... Surjective Operators . . . . . . . . . . . . . . . . . . . . . 46. Comments on Chapter 2 ...
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[PDF] Lectures in Functional Analysis Roman Vershynin - UCI Mathematics
Minor borrowings are made from Yoshida “Functional analysis”, Rudin “Functional analysis”, and Conway “A course in functional analysis”. For some topics not ...
[20]
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[PDF] Open Mapping Theorem
The Open Mapping Theorem Let X and Y be Banach spaces and T : X -+ Y a bounded linear transformation mapping X onto Y. Then Tis an open mapping, i.e. TU is ...
[22]
Show that the Open Mapping Theorem requires both spaces to be ...
Dec 9, 2014 · For a counterexample with incomplete X you can use e.g. a Hamel basis of ℓ1 to produce a discontinuous linear functional f:ℓ1→R. Then define ...Missing: multiplication | Show results with:multiplication
[23]
Theorie Des Operations Lineaires : Banach Stefan - Internet Archive
Jan 24, 2017 · Theorie Des Operations Lineaires Book Source: Digital Library of India Item 2015.493033 dc.contributor.author: Banach StefanMissing: mapping statement
[24]
[PDF] FUNCTIONAL ANALYSIS - ETH Zürich
Jun 8, 2017 · These are notes for the lecture course “Functional Analysis I” held by the second author at ETH Zürich in the fall semester 2015.
[25]
[PDF] Use of Functional Analysis in Solving Equations 1 - IIT Madras
We shall characterize well-posedness of the equation (1.1) for the case when A is a closed operator. We may recall that, if X and Y are normed linear spaces and ...
[26]
The closed graph theorem in various categories - Terry Tao
Nov 20, 2012 · The closed graph theorem is a corollary of the open mapping theorem, which asserts that any continuous linear surjection from one Banach space to another is ...
[27]
proof of closed graph theorem - PlanetMath
Mar 22, 2013 · Let T:X→Y T : X → Y be a linear mapping. Denote its graph by G(T) G ⁢ ( T ) , and let p1:X×Y→X p 1 : X × Y → X and p2:X×Y→Y p 2 : X × Y → Y ...Missing: statement | Show results with:statement
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A quick application of the closed graph theorem - Terry Tao
Apr 22, 2016 · If E is an infinite dimensional Banach space, then there exists a discontinuous linear functional φ: E → R. Take u ∈ E such that φ(u) = 1 ...
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An example of a non closed linear map. - Math Stack Exchange
Jan 7, 2022 · Just take any non-continuous linear map on a Banach space, an example of a non-continuous functional (so image R) can be found here.Example of a linear operator whose graph is not closed but it takes a ...discontinuous Linear operator from a Banach space to a normed ...More results from math.stackexchange.com
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[PDF] FUNCTIONAL ANALYSIS | Second Edition Walter Rudin
Functional analysis/Walter Rudin.-2nd ed. p. em. -(international series in ... open mapping theorem. 47. The closed graph theorem. 50. Bilinear mappings. 52.
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[PDF] arXiv:2204.10015v1 [math.FA] 21 Apr 2022
Apr 21, 2022 · The open mapping theorem guarantees that this definition is equivalent to the usual ... exact sequences of Banach spaces: if. (8). 0. // Y ı. // X.
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Stefan Banach (1892 - 1945) - Biography - University of St Andrews
Banach founded modern functional analysis and made major contributions to the theory of topological vector spaces. In addition, he contributed to measure theory ...Missing: origins | Show results with:origins
[33]
quantum theory: von Neumann vs. Dirac
Jul 27, 2004 · In the mid-1930s, he worked extensively on lattice theory (see the entry on quantum logic), rings of operators, and continuous geometries.
[34]
[PDF] The Baire category theorem
Proof of the strong version of the open mapping theorem. Let T be a bounded linear map from a Banach space X into a normed linear space Y . If T[X] is ...
[35]
[PDF] some remarks on the history of functional analysis
... Banach Open Map- ping Theorem. It is by many authorities regarded as one of three fundamental. “pillars” of the functional analysis. In the paper [3] there ...
[36]
The Fixed Point Theorems of Brouwer and Schauder - SpringerLink
We are going to dedicate the first chapter to the study of the fixed point theorem of Schauder [S, 1930]. We have divided the chapter into two parts: In the ...
[37]
[PDF] On the origin and early history of functional analysis - DiVA portal
Between. 1920 and 1932, both Banach and Hans Hahn (1879 – 1934) published their books on the subject which contained the major theorems of functional analysis; ...
[38]
A strong open mapping theorem for surjections from cones onto ...
Feb 12, 2013 · This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael's Selection Theorem to yield the ...Missing: alternative proof
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Topological Vector Spaces: Chapters 1–5 - SpringerLink
This is a softcover reprint of the English translation of 1987 of the second edition of Bourbaki's Espaces Vectoriels Topologiques (1981).Missing: mapping | Show results with:mapping
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[PDF] Learning Maps Between Function Spaces With Applications to PDEs
We prove a universal approximation theorem for our proposed neural operator, showing that it can approximate any given nonlinear continuous operator.
[41]
[PDF] frechet.pdf - Fréchet spaces - Purdue Math
Mar 3, 2025 · Theorem 2 (Open Mapping Theorem). If X and Y are Fréchet spaces, and if T : X → Y is a surjective continuous linear function, then T(U) ⊂ ...Missing: standard book
[42]
Barrelled spaces and the open mapping theorem - SpringerLink
Aug 27, 2006 · Barrelled spaces and the open mapping theorem. In: Topological Vector Spaces. Lecture Notes in Mathematics, vol 639. Springer, Berlin ...