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Bounded operator

In , a bounded operator (or bounded linear operator) is a T: V \to W between normed vector spaces V and W such that there exists a constant C > 0 with \|Tv\|_W \leq C \|v\|_V for all v \in V. This condition ensures that T is continuous and preserves the boundedness of sets, making it a fundamental concept for studying linear transformations in infinite-dimensional spaces. Bounded operators form a B(V, W) under pointwise addition and , equipped with the \|T\| = \sup_{\|v\|_V \leq 1} \|Tv\|_W, which quantifies the maximum "stretching" effect of T and coincides with the smallest such constant C. If W is a , then B(V, W) is itself a with respect to this norm, enabling the application of completeness arguments in . Key properties include the fact that boundedness is equivalent to for linear operators on normed spaces, and the (or Banach-Steinhaus theorem) provides conditions under which pointwise bounded families of operators are uniformly bounded. Examples of bounded operators abound in , such as operators on spaces like \ell_p (where the multiplier is bounded), integral operators Tf(x) = \int k(x,y) f(y) \, dy on L_p spaces with continuous kernels k, and differential operators when suitably interpreted on Sobolev spaces. These operators are central to , where the spectrum of a bounded operator on a includes eigenvalues and approximate eigenvalues, and to the study of compact operators, operators, and the , which links bounded linear functionals to dual spaces. In Hilbert spaces, important subclasses of bounded operators, such as operators, facilitate applications in and partial differential equations.

Normed and Banach Spaces

Definition in Normed Spaces

In a normed vector space, the concept of boundedness for linear operators provides a measure of their controlled growth relative to the underlying norms. Consider two normed vector spaces (X, \|\cdot\|_X) and (Y, \|\cdot\|_Y). A linear operator T: X \to Y is bounded if there exists a constant M \geq 0 such that \|Tx\|_Y \leq M \|x\|_X for all x \in X. Equivalently, T is bounded if \sup_{\|x\|_X \leq 1} \|Tx\|_Y < \infty. This definition applies without requiring completeness of the spaces, distinguishing it from settings in Banach spaces. The operator norm of a bounded linear operator T is defined as \|T\| = \sup_{\|x\|_X \leq 1} \|Tx\|_Y, which equals the infimum of all such constants M satisfying the boundedness inequality. This norm satisfies \|Tx\|_Y \leq \|T\| \|x\|_X for all x \in X. For compositions of bounded operators, submultiplicativity holds: if S: Y \to Z is another bounded linear operator on a normed space Z, then \|S \circ T\| \leq \|S\| \|T\|. These properties make the space of bounded operators B(X, Y) itself a normed space under the operator norm. Simple examples illustrate boundedness in normed spaces. The scalar multiplication operator T: X \to X given by Tx = \lambda x for fixed \lambda \in \mathbb{C} (or \mathbb{R}) is bounded with \|T\| = |\lambda|. Another example is the projection onto a finite-dimensional subspace; for instance, in \ell^p spaces with $1 \leq p < \infty, the coordinate projection \pi_k((x_j)_{j=1}^\infty) = x_k onto the one-dimensional subspace spanned by the k-th standard basis vector is bounded with \|\pi_k\| = 1. Such projections extend to higher finite dimensions while remaining bounded, as all linear operators on finite-dimensional normed spaces are bounded.

Equivalence to Continuity

In normed linear spaces X and Y, a linear operator T: X \to Y is bounded if and only if it is continuous at every point in X, or equivalently, continuous at the origin $0 \in X. This equivalence holds because linearity ensures that continuity at one point implies continuity everywhere, and boundedness provides a uniform control on the operator's growth relative to the norms. To see that boundedness implies continuity, suppose \|T\| < \infty is the operator norm of T. Then, for any x, y \in X, \|Tx - Ty\|_Y = \|T(x - y)\|_Y \leq \|T\| \cdot \|x - y\|_X. This shows that T is Lipschitz continuous with constant \|T\|, hence uniformly continuous on X. Conversely, if T is continuous at $0, there exists \delta > 0 such that \|x\|_X < \delta implies \|Tx\|_Y < 1. For arbitrary x \in X with x \neq 0, set z = (\delta / \|x\|_X) x, so \|z\|_X = \delta and \|Tz\|_Y < 1. By linearity, \|Tx\|_Y = \frac{\|x\|_X}{\delta} \|Tz\|_Y < \frac{\|x\|_X}{\delta}, establishing boundedness with constant M = 1/\delta. This demonstrates that continuity at $0 implies boundedness on the unit ball of X, and thus globally. The equivalence bridges topology and analysis by showing that all continuous linear operators on normed spaces are automatically bounded, enabling the operator norm to serve as a precise measure of continuity's "strength." This result is foundational, as it allows topological properties like compactness or convergence to be analyzed through analytic tools such as norm estimates. This equivalence highlights the role of norms in making abstract continuity concrete, a development central to functional analysis since its early days. Key early proofs appeared in the works of Fréchet (1906) under separability assumptions and were fully generalized by Banach in 1932, solidifying the framework for normed spaces.

Relative Boundedness

In functional analysis, an operator T defined on a dense linear subspace of a normed space X is said to be A-bounded, or relatively bounded with respect to another densely defined operator A: D(A) \to X, if D(A) \subset D(T) and there exist constants a, b \geq 0 such that \|Tx\| \leq a \|Ax\| + b \|x\| for all x \in D(A). The infimum of all possible values of a over such inequalities is called the relative bound (or A-bound) of T. This concept extends the notion of absolute boundedness by allowing perturbations controlled relative to the "size" of A x, which is particularly useful when T is unbounded in the usual sense but behaves well compared to A. If b = 0, the inequality simplifies to \|Tx\| \leq a \|Ax\|, and T is called absolutely A-bounded (or A-continuous). In this case, T extends continuously to the completion of D(A) under the graph norm \|x\|_A = \|x\| + \|Ax\|, making T bounded with respect to this stronger norm topology. More generally, relative boundedness implies that T is continuous from (D(A), \|\cdot\|_A) to X, providing a framework for analyzing perturbations on incomplete or non-dense domains. Relative boundedness plays a central role in perturbation theory for linear operators, particularly in determining closability, extendability, and essential self-adjointness of sums A + T. For instance, if A is closed and densely defined, and T is A-bounded with relative bound a < 1 and b sufficiently small, then A + T is closed and densely defined, as established by the . This theorem underpins stability results for differential operators under perturbations, ensuring that spectral properties are preserved under suitable relative controls. A representative example arises in the study of differential operators on L^2(\mathbb{R}), where the differentiation operator A = \frac{d}{dx} is defined on the Sobolev space H^1(\mathbb{R}). The multiplication operator T f = x f (position operator) is A-bounded, satisfying \|x f\| \leq a \|f'\| + b \|f\| for suitable a, b > 0 and all f \in H^1(\mathbb{R}), which follows from or showing the relative control. This illustrates how multiplication operators can perturb while remaining relatively bounded on appropriate domains.

Uniform Boundedness Principle

The uniform boundedness principle, also known as the Banach–Steinhaus theorem, asserts that if X is a Banach space, Y is a normed space, and \{T_\alpha : X \to Y\}_{\alpha \in A} is a family of bounded linear operators such that \sup_{\alpha \in A} \|T_\alpha x\| < \infty for every x \in X, then \sup_{\alpha \in A} \|T_\alpha\| < \infty. This result establishes that pointwise boundedness of the family implies uniform boundedness in the operator norm. The completeness of the domain space X is essential for the principle to hold, as it fails in incomplete normed spaces. For instance, consider the space c_{00} of sequences with finitely many nonzero terms, equipped with the \ell^\infty norm, which is incomplete; the family of right-shift operators T_n defined by (T_n x)_k = x_{k-n} for k \geq n and 0 otherwise is pointwise bounded but not uniformly bounded, since \|T_n\| = 1 for the standard basis vector but grows unboundedly on certain elements. A standard proof proceeds via the Baire category theorem applied to the unit ball of X. Define E_n = \{x \in X : \sup_\alpha \|T_\alpha x\| \leq n\} for n \in \mathbb{N}; each E_n is closed, and their union covers X by pointwise boundedness. Thus, some E_n has nonempty interior, say containing a ball B(x_0, r) with r > 0, implying that the family is uniformly bounded on ball and hence everywhere. An alternative proof uses the contrapositive and the closed graph theorem: if \sup_\alpha \|T_\alpha\| = \infty, then there exists a sequence \alpha_k with \|T_{\alpha_k}\| \to \infty, and by considering the graph of the operator mapping to the , one derives a to pointwise boundedness in incomplete cases, but the full argument relies on . The principle yields several important corollaries in the theory of Banach spaces. The closed graph theorem follows by applying the principle to the family of operators S_y(x) = \frac{Tx - y}{\|x\| + 1} for y \in Y, showing that a densely defined closed linear operator between Banach spaces extends to a bounded . Similarly, the open mapping theorem—that a surjective bounded linear operator between Banach spaces maps open sets to open sets—derives from the principle by considering the family of finite-rank projections onto the range, ensuring uniform boundedness implies openness. These equivalences highlight the principle's foundational role in .

Hilbert Spaces

Bounded Operators

In the context of , a linear T: H \to K between H and K is bounded if there exists a constant M \geq 0 such that \|Tx\| \leq M \|x\| for all x \in H. The smallest such M, denoted \|T\|, is called the of T. This definition aligns with the general normed space case but leverages the inner product structure of , where the norm is induced by the inner product \langle \cdot, \cdot \rangle, ensuring that boundedness implies uniform control over the output in the . Bounded linear operators on Hilbert spaces are defined on the entire H, with no initial domain restrictions, distinguishing them from potentially unbounded operators that require dense subspaces for definition. They map bounded sets in H to bounded sets in K, preserving the structure of bounded regions under the operator action. The of two bounded operators T: H \to K and S: K \to L is bounded, with \|ST\| \leq \|S\| \|T\|, and scalar multiples satisfy \|\alpha T\| = |\alpha| \|T\| for \alpha \in \mathbb{C} (or \mathbb{R}). As in normed spaces, boundedness is equivalent to . A classic example of a bounded operator on the L^2[a, b] is an with a continuous k(t, s) on the compact interval [a, b] \times [a, b], defined by (Tf)(t) = \int_a^b k(t, s) f(s) \, ds for f \in L^2[a, b]. The of k ensures T is bounded, with controlled by the maximum of |k(t, s)|, and the operator maps the complete space L^2[a, b] into itself while preserving the completeness of the space. Another prominent example is the \mathcal{F} on the L^2(\mathbb{R}), given by (\mathcal{F} f)(\xi) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) e^{-i x \xi} \, dx, extended by density from Schwartz functions. By the , \mathcal{F} is a bounded linear (in fact, unitary) with \|\mathcal{F} f\|_{L^2} = \|f\|_{L^2} for all f \in L^2(\mathbb{R}), thus maintaining the structure and completeness.

Adjoints and Self-Adjoint Operators

In Hilbert spaces, the adjoint of a bounded linear provides a fundamental duality structure. For bounded operators T: H \to K between Hilbert spaces H and K, the T^*: K \to H is the unique bounded linear satisfying \langle Tx, y \rangle_K = \langle x, T^* y \rangle_H for all x \in H and y \in K, where \langle \cdot, \cdot \rangle_H and \langle \cdot, \cdot \rangle_K denote the respective inner products. This exists and is unique by the applied to the induced by T. Moreover, the satisfies \|T^*\| = \|T\|, as the equality follows from the and bounding the norms in both directions. Key properties of the include and a to the square of the : (T^*)^* = T and \|T^* T\| = \|T\|^2. The property arises from applying the defining equation twice, confirming that the of the recovers the original operator. The relation \|T^* T\| = \|T\|^2 holds because T^* T is positive semi-definite, with its equaling the square of T's via the inner product \langle T^* T x, x \rangle = \|T x\|^2. A bounded operator T: H \to H on a H is if T = T^*, meaning \langle T x, y \rangle = \langle x, T y \rangle for all x, y \in H. operators have real , as eigenvalues \lambda satisfy \langle T x, x \rangle = \lambda \|x\|^2 for eigenvectors x, implying \lambda = \overline{\lambda} by the property. By the , every bounded operator on a separable is unitarily equivalent to a operator by a bounded real-valued function on L^2(\mu) for some , providing a functional model for its action. Representative examples of self-adjoint operators include and multiplication operators by bounded real functions. An P: H \to H onto a closed satisfies P^2 = P and P = P^*, as \langle P x, y \rangle = \langle x, P y \rangle follows from the into and . For instance, the projection onto even functions in L^2(\mathbb{R}) given by P f(x) = \frac{f(x) + f(-x)}{2} is . Similarly, the multiplication M_m on L^2([0,1]) defined by (M_m f)(x) = m(x) f(x), where m \in L^\infty([0,1]) is real-valued, is because \langle M_m f, g \rangle = \int m |f|^2 = \langle f, M_m g \rangle.

Topological Vector Spaces

Bounded Sets and Operators

In topological vector spaces, the concept of boundedness generalizes the notion from normed spaces by relying on the absorption property rather than a single . A B of a Y is called bounded if, for every neighborhood U of the in Y, there exists a scalar \lambda > 0 such that \lambda B \subset U. This definition captures sets that can be "scaled down" to fit within any given neighborhood of zero, reflecting the topological structure without presupposing a . A linear T: X \to Y between topological spaces X and Y is bounded if it maps bounded subsets of X to bounded subsets of Y. Equivalently, T is bounded if the image under T of any set absorbed by a neighborhood of zero in X is absorbed by some neighborhood of zero in Y. This property ensures that T preserves the "size" of sets in a topological sense, independent of metric considerations. In locally convex topological vector spaces, where the topology is generated by a separating family of seminorms \{p_i\}_{i \in I}, boundedness of a set B is characterized by the condition that \sup_{b \in B} p_i(b) < \infty for every seminorm p_i. This ties bounded sets directly to the uniform structure induced by the seminorms, as the entourages of the uniformity are defined via finite collections of these seminorms, allowing bounded sets to be uniformly controlled across the family. Unlike in normed spaces, where boundedness of a linear operator is equivalent to continuity, this equivalence fails in non-normable topological vector spaces. While every continuous linear operator maps bounded sets to bounded sets, a bounded linear operator need not be continuous in such spaces. For instance, continuity requires preservation of the topology at every point, but boundedness only constrains behavior on bounded sets, which may not suffice without additional structure like metrizability.

Continuity and Boundedness

In topological vector spaces, the concepts of continuity and boundedness for linear operators are closely related but not always equivalent. A linear operator T: X \to Y between topological vector spaces is bounded if it maps every bounded subset of X to a bounded subset of Y, where a subset is bounded if it is absorbed by every neighborhood of the origin. Every continuous linear operator between topological vector spaces is bounded. To see this, suppose T is continuous at the origin. Let B \subseteq X be bounded and U a neighborhood of the origin in Y. There exists a neighborhood V \subseteq X of the origin such that T(V) \subseteq U. Since B is bounded, there is a scalar \lambda > 0 such that B \subseteq \lambda V, so T(B) \subseteq \lambda U, implying T(B) is bounded. The converse—that every is continuous—does not hold in general topological vector spaces. However, it holds if the domain X is bornological, meaning that every , balanced, absorbing set that contains a multiple of every is a neighborhood of the . In such spaces, for a T: X \to Y and a neighborhood U of the in Y, the preimage T^{-1}(U) absorbs every in X because T maps to , which are absorbed by U. By the bornological property, there exists a , balanced neighborhood W of the in X contained in T^{-1}(U), so T(W) \subseteq U, proving continuity at the . Counterexamples arise in non-bornological topological vector spaces, such as certain inductive limits of that fail to be bornological. In these settings, there exist linear operators that map bounded sets to bounded sets but fail to be continuous, as the does not ensure that preimages of neighborhoods absorb bounded sets in a neighborhood-generating manner. For instance, the identity operator from an infinite-dimensional equipped with its to the same space with its norm is bounded—by Mackey's theorem, weakly bounded sets are norm-bounded—but discontinuous, since the is strictly coarser than the norm . In general topological vector spaces, implies boundedness, making the stronger property, while boundedness is weaker since there are bounded operators that are discontinuous. This divergence highlights the role of additional structural assumptions, like bornologicality, for equivalence. Conversely, continuous linear operators always map compact sets to bounded sets, as compact sets are bounded and images under continuous linear maps remain compact in Hausdorff spaces, hence bounded.

Bornological and Mackey Spaces

In bornological spaces, a class of topological vector spaces (TVS), the notions of boundedness and continuity for linear operators coincide. Specifically, a locally convex TVS is bornological if every bornivore is a neighborhood of the origin, where a bornivore is a balanced that absorbs every . This property ensures that a linear operator between bornological spaces is continuous if and only if it maps to . Normed spaces provide a fundamental example of bornological spaces. Mackey spaces represent another important class of locally convex TVS where boundedness relates closely to continuity, particularly through the dual structure. A Mackey space is a locally convex TVS equipped with the , defined as the strongest locally convex compatible with its continuous dual. In such spaces, every weakly bounded set (bounded in the ) is strongly bounded, ensuring that continuous linear operators map to bounded sets in a manner consistent with the topology's minimality relative to the dual. Examples of Mackey spaces include all normed spaces. While many Fréchet spaces are Mackey spaces, not all are, as there exist Fréchet topologies strictly coarser than the . The bornological topology is generally coarser than the Mackey topology on a given locally convex space, reflecting differences in how they prioritize bounded sets versus dual continuity. This distinction proves useful in constructions like inductive limits, where inductive limits of bornological spaces remain bornological, facilitating the study of operators on spaces arising as unions of increasing sequences of normed spaces.

Characterizations of Bounded Operators

In topological vector spaces, a linear operator T: X \to Y is defined to be bounded if it maps every bounded subset of X to a bounded subset of Y. This characterization extends the notion from normed spaces, where boundedness corresponds to the existence of a uniform bound on the operator norm, but in general topological vector spaces, it relies on the absorption property of bounded sets by neighborhoods of the origin. Continuous linear operators are always bounded, as the image of a bounded set under a continuous map remains bounded. An equivalent sequential characterization holds: T is bounded if and only if it maps every sequentially bounded subset of X to a sequentially bounded subset of Y. A subset is sequentially bounded if, for every continuous linear functional, the images of its sequences are bounded in the . This formulation is particularly useful in sequentially complete spaces, where sequential aligns more closely with boundedness properties. Sequentially continuous linear operators are bounded, though the converse requires the domain to be a . Another intrinsic characterization is that T is bounded if and only if it is uniformly continuous on every bounded subset of X. Uniform continuity on a bounded set means that for every neighborhood V of the origin in Y, there exists a neighborhood U of the origin in X such that if A \subset X is bounded and x, y \in A with x - y \in U, then T(x) - T(y) \in V. This property generalizes the uniform boundedness seen in normed spaces and facilitates applications in uniform structures on topological vector spaces. In locally convex topological vector spaces, boundedness of T is equivalent to continuity restricted to a neighborhood basis at the origin. Specifically, T is continuous (and hence bounded) if and only if for some (equivalently, every) balanced neighborhood U of the origin in X, the image T(U) is bounded in Y. This leverages the seminorm structure defining the , where bounded sets are those with finite values, ensuring the equivalence holds without additional assumptions on the space. In complete topological vector spaces, boundedness relates to the closed graph property under mild assumptions, such as metrizability of the topology. For instance, if X is a metrizable (an F-space) and T: X \to Y has a closed graph, then T is continuous and thus bounded. This connection via the closed graph theorem provides a way to verify boundedness through graphical conditions, though it requires to ensure the graph's implies continuity. Unlike the case, where boundedness admits a \|T\| = \sup_{\|x\| \leq 1} \|T x\|, general topological vector spaces lack such a single quantifying , relying instead on set-theoretic boundedness. This contrast highlights limitations in operator and without additional structure like local boundedness.

Operator Algebras and Extensions

Bounded Operators in

In the multiplicative framework of , bounded operators are typically understood as bounded homomorphisms, which are linear maps between that preserve the multiplication operation while respecting the structure. Specifically, given A and B, a bounded homomorphism \phi: A \to B is a continuous linear map satisfying \phi(ab) = \phi(a)\phi(b) for all a, b \in A, and \phi(e_A) = e_B if both algebras are unital with identities e_A and e_B. Such maps are norm-bounded, with the \|\phi\| = \sup_{\|a\| \leq 1} \|\phi(a)\|, ensuring compatibility with the submultiplicative of the algebras. For endomorphisms on a single Banach algebra A, the relevant bounded operators are the multipliers, which form the multiplier algebra M(A). A left multiplier is a bounded linear map T: A \to A such that T(ab) = T(a)b for all a, b \in A, while a right multiplier satisfies T(ab) = a T(b); the two-sided multipliers are those that are both. The space M(A) of all two-sided multipliers is itself a Banach algebra under pointwise addition and scalar multiplication, with multiplication defined by composition (i.e., (T_1 T_2)(a) = T_1(T_2(a))), and equipped with the operator norm, making it complete and submultiplicative. This structure extends the operator algebra B(A) of all bounded linear operators on the underlying Banach space, but restricts to those preserving the algebra multiplication. In the commutative setting, Gelfand theory provides a concrete realization of bounded multipliers through the spectrum. For a commutative unital A, the Gelfand \Delta(A) is the space of nonzero multiplicative linear functionals (characters) on A, which are automatically bounded. The Gelfand transform extends to multipliers, identifying M(A) isometrically with C(\Delta(A)), the Banach algebra of continuous complex-valued functions on \Delta(A) under the sup norm; each bounded multiplier corresponds to a bounded continuous function on the via this isomorphism. A prominent example arises in the commutative C(K) of continuous functions on a compact K, equipped with and the supremum \|f\|_\infty = \sup_{k \in K} |f(k)|. Here, the left and right multipliers coincide due to commutativity and are precisely the operators M_g: f \mapsto gf for g \in C(K), forming M(C(K)) = C(K) as a unital . This illustrates how bounded multipliers in such algebras recover the original structure through fixed function multiplications.

Bounded Operators in C*-Algebras

C*-algebras arise naturally as norm-closed subalgebras of the algebra of bounded linear operators on a that are closed under the operation. Abstractly, a is a complex equipped with a satisfying the C*-identity \|a^* a\| = \|a\|^2 for all elements a. This , known as the C*-norm, can be characterized representation-theoretically as \|a\| = \sup \{\|\pi(a)\| : \pi \text{ is a non-degenerate *-representation of } A \text{ on a Hilbert space}\}, where the supremum is taken over all faithful representations \pi. Bounded -homomorphisms between C-algebras are contractive, meaning they preserve the C*-norm up to a factor of at most 1, and become isometric when the kernel is zero. For normal elements a in a C*-algebra, where a a^* = a^* a, the spectral radius equals the norm: r(a) = \|a\|. The C*-subalgebra generated by a normal element a is the uniform closure of the polynomials in a and a^*. Non-degenerate *-representations of on map to bounded operators, and by the Gelfand-Naimark theorem, every admits a faithful as a closed *-subalgebra of bounded operators on some . Conversely, the generated by any set of bounded operators on a , closed under adjoints and uniform limits, forms a . In this structure, positive elements—those expressible as b^* b for some b—are and bounded, with spectra contained in the non-negative reals, ensuring the positivity condition aligns with the bounded operators from theory.

Examples and Applications

Bounded Linear Operators

In normed linear spaces, the identity I, defined by Ix = x for all x in the space, exemplifies a bounded linear with \|I\| = 1, since \|Ix\| = \|x\| \leq 1 \cdot \|x\| holds for the induced . Finite-rank operators, which map into a finite-dimensional , provide another class of bounded operators; for instance, a rank-one of the form Tx = \langle x, y \rangle z (where \langle \cdot, \cdot \rangle denotes the inner product in a Hilbert space) satisfies \|Tx\| \leq \|y\| \|z\| \|x\|, yielding \|T\| \leq \|y\| \|z\|. More generally, any finite-rank , as a finite sum of rank-one operators, inherits boundedness from this property. A concrete example in Hilbert spaces is the Volterra operator V on L^2[0,1], defined by (Vf)(x) = \int_0^x f(t) \, dt for f \in L^2[0,1]. This operator is bounded, with \|Vf\|_2 \leq \|f\|_2 implying \|V\| \leq 1, though the exact norm is \|V\| = 2/\pi \approx 0.637. To verify boundedness, note that by the Cauchy-Schwarz inequality, \|(Vf)(x)\|^2 \leq x \int_0^x |f(t)|^2 \, dt \leq \int_0^1 |f(t)|^2 \, dt, and integrating over x yields the L^2 bound. Unlike unbounded counterparts such as differentiation, which cannot be defined on the full space without restrictions, the Volterra operator acts on the entire L^2[0,1] due to the smoothing effect of integration. In Banach spaces like L^1(\mathbb{R}), convolution operators T_g f = f * \mu, where \mu is a bounded measure with total variation \|\mu\| < \infty, are bounded linear operators with \|T_g\| \leq \|\mu\|. This follows from \|f * \mu\|_1 \leq \|f\|_1 \|\mu\|, a consequence of Fubini's theorem and the definition of . A related class consists of integral operators Tf(x) = \int K(x,y) f(y) \, dy on spaces like L^2[a,b], where the kernel K is continuous on the compact set [a,b] \times [a,b]; such operators are bounded because \|Tf(x)\| \leq \left( \int |K(x,y)| \, dy \right) \|f\|_\infty \leq \sup_x \int |K(x,y)| \, dy \cdot \|f\|, and continuity ensures the supremum is finite. These examples highlight how regularity of the kernel or measure guarantees definition and boundedness on the whole space, in contrast to irregular kernels that may lead to unboundedness.

Unbounded Linear Operators

In functional analysis, an unbounded linear operator between normed vector spaces X and Y is a linear map T: D(T) \to Y, where D(T) is a proper dense linear subspace of X, such that there exists no constant M > 0 satisfying \|Tx\| \leq M \|x\| for all x \in D(T). This contrasts with bounded operators, which extend continuously to the entire space X. Unbounded operators arise naturally when the domain must be restricted to ensure the map is well-defined, as extending to the full space would violate linearity or . A classic example is the operator T x = x' defined on the of infinitely differentiable functions D(T) = C^\infty[0,1] with the supremum on C[0,1]. To see that T is unbounded, consider the sequence x_n(t) = \sin(2\pi n t); then \|x_n\|_\infty = 1, but \|x_n'\|_\infty = 2\pi n \to \infty as n \to \infty, so \sup_{x \neq 0} \|Tx\| / \|x\| = \infty. Similarly, in Hilbert spaces, the P = -i \frac{d}{dx} acts on L^2(\mathbb{R}) with domain the of smooth functions with compact D(P) = C_c^\infty(\mathbb{R}); it is unbounded because functions with increasingly rapid oscillations, such as \phi_n(x) = n^{1/2} \psi(n x) for a fixed test function \psi, satisfy \|\phi_n\|_2 = 1 while \|P \phi_n\|_2 \approx n \to \infty. Unbounded operators are often not closed, but many are closable, meaning their graph \Gamma(T) = \{(x, Tx) \mid x \in D(T)\} has a closure that is the graph of another linear operator, allowing a minimal closed extension \overline{T}. For instance, the differentiation operator on C^1[0,1] with boundary conditions x(0) = x(1) = 0 is closable, extending to the Sobolev space W_0^{1,p}[0,1] for $1 < p < \infty. An operator T is closable if and only if its adjoint T^* is densely defined. The completeness of the ambient X (e.g., as a ) does not prevent unboundedness, since the issue lies in the restricted D(T), where no uniform bound holds despite D(T) being dense and the linear on it; for example, C^\infty[0,1] is complete in the Fréchet topology, yet differentiation remains unbounded.

Applications in Analysis and Physics

In partial differential equations (PDEs), elliptic operators play a crucial role in establishing the existence and uniqueness of weak solutions for boundary value problems. For instance, the with Dirichlet boundary conditions on a bounded generates a bounded bilinear form that satisfies the coercivity and continuity conditions required by the Lax-Milgram theorem, thereby guaranteeing well-posedness in Hilbert spaces like Sobolev spaces. This theorem underpins variational methods for solving such PDEs. In , bounded operators represent observables that are well-defined on the entire of states, such as approximations to or operators confined to bounded regions. In contrast, fundamental operators like the are typically unbounded, but bounded perturbations or resolvents facilitate and . Stone's theorem establishes a between strongly continuous one-parameter unitary groups—essential for in —and self-adjoint (possibly unbounded) generators, with bounded operators arising in the context of the generator's resolvent. This framework ensures the unitarity of time-evolution operators generated by bounded self-adjoint perturbations. Bounded projections onto spline spaces are vital in approximation theory for , enabling stable and efficient of functions in finite element methods. These projections, such as L²-projections onto spline subspaces, maintain bounded norms independent of refinement, allowing for optimal estimates in approximating functions over bounded intervals. Seminal estimates for such projections highlight their role in Ritz-Galerkin approximations for elliptic PDEs, where boundedness ensures convergence rates dictated by the spline degree. In more advanced settings, bounded operators are instrumental in Fréchet spaces for handling infinite-dimensional PDEs, where the complete metrizable topology supports continuity without a single . For example, in spaces of smooth functions with rapidly decreasing seminorms, bounded linear operators preserve solutions to evolution equations, facilitating well-posedness in non-Hilbertian frameworks like those for Navier-Stokes equations. Complementing this, LF-spaces—inductive limits of Fréchet spaces—model distributions in , where bounded operators extend differential operators continuously, enabling the study of generalized solutions to PDEs involving singular data, as in the theory of tempered distributions.