DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-01

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A bounded linear operator between normed spaces

Definition

Let $X$ and $Y$ be normed spaces over the same scalar field $K$ , read in the real case from A norm on a real vector space, the induced metric, and the dictionary with the metric axioms and in the complex case from Real and complex scalar conventions for normed spaces. A linear map $T : X \to Y$ (Linear map between vector spaces over the same field) is a bounded linear operator when there is a real constant $C \geq 0$ such that

$∥ T x ∥ \leq C ∥ x ∥ for all x \in X .$

Any such $C$ is called a bound for $T$ .

Remarks

The zero operator is bounded with bound $0$ .
A bound is not unique: if $C$ works and $C^{'} \geq C$ , then $C^{'}$ works as well.
When $X = {0}$ , every linear map $X \to Y$ is bounded with bound $0$ .

Depends on

Used by

A bounded map into H¹₀ yields a compact L² operator Corollary
A linear map from a finite-dimensional normed space is bounded Corollary
A one-sided comparison of two complete norms makes them equivalent Corollary
A pointwise limit of bounded operators is bounded with the liminf norm bound Corollary
A semigroup with unbounded generator is not norm continuous at zero Corollary
Abstract parabolic smoothing for mild solutions Corollary
Analytic semigroups are operator-norm differentiable away from zero Corollary
Compact operator iff approximation numbers tend to zero Corollary
Finite rank operators are norm dense in compact Hilbert space operators Corollary
Lambda identity minus compact has index zero Corollary
On an infinite-dimensional normed space, the identity operator is not compact Corollary
Quadratic spectral bounds control a self-adjoint parabolic semigroup Corollary
The Lax--Milgram solution operator has norm at most 1/α Corollary
Uniqueness implies existence for the elliptic Dirichlet problem Corollary
A bounded bijection of incomplete normed spaces need not be open Counterexample
A bounded form without coercivity need not be solvable Counterexample
A closed subspace of ell-infinity that is not complemented Counterexample
A compact operator can have nondense range Counterexample
A complete domain is necessary for sequential uniform boundedness Counterexample
A first resolvent estimate does not ensure the prescribed semigroup bound Counterexample
A strongly continuous unitary group need not be norm continuous Counterexample
An analytic semigroup need not be norm continuous at zero Counterexample
Compactness is not preserved by strong operator limits Counterexample
Identity is compact iff the space is finite dimensional Counterexample
Point evaluation is unbounded below the Sobolev continuity threshold Counterexample
Strong continuity does not imply operator-norm continuity Counterexample
The sector changes under the sign convention Counterexample
The trace estimate fails on an outward cusp above the critical sharpness Counterexample
The unilateral shift obstructs a cyclic linear trace extension Counterexample
Uniform boundedness fails on the incomplete space c₀0 Counterexample
A bounded operator that is bounded below Definition
A complemented closed subspace of a normed space Definition
A topological isomorphism of normed spaces Definition
Absolute value and singular values of a compact operator Definition
Adjoint of a densely defined operator Definition
Approximable operator Definition
Approximation property and bounded approximation property Definition
Bounded left inverses and bounded right inverses Definition
C star algebra Definition
Cayley transform of a self-adjoint operator Definition

…and 172 more results.

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Theo Buhler and Dietmar A. Salamon, Functional Analysis (standard reference, not scraped)
Gerald Teschl, Topics in Real and Functional Analysis (standard reference, not scraped)