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In functional analysis and operator theory, a **bounded linear operator** is a linear transformation between topological vector spaces (TVSs) and that maps bounded subsets of to bounded subsets of
If and are normed vector spaces (a special type of TVS), then is bounded if and only if there exists some such that for all
The smallest such is called the operator norm of and denoted by
A bounded operator between normed spaces is continuous and vice versa.

The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces.

Outside of functional analysis, when a function is called "bounded" then this usually means that its image is a bounded subset of its codomain. A linear map has this property if and only if it is identically Consequently, in functional analysis, when a linear operator is called "bounded" then it is never meant in this abstract sense (of having a bounded image).

Every bounded operator is Lipschitz continuous at

A linear operator between normed spaces is bounded if and only if it is continuous.

Suppose that is bounded. Then, for all vectors with nonzero we have Letting go to zero shows that is continuous at Moreover, since the constant does not depend on this shows that in fact is uniformly continuous, and even Lipschitz continuous.

Conversely, it follows from the continuity at the zero vector that there exists a such that for all vectors with Thus, for all non-zero one has This proves that is bounded. Q.E.D.

A linear operator between two topological vector spaces (TVSs) is called a ** bounded linear operator** or just

Every sequentially continuous linear operator between TVS is a bounded operator.^{[1]}
This implies that every continuous linear operator between metrizable TVS is bounded.
However, in general, a bounded linear operator between two TVSs need not be continuous.

This formulation allows one to define bounded operators between general topological vector spaces as an operator which takes bounded sets to bounded sets. In this context, it is still true that every continuous map is bounded, however the converse fails; a bounded operator need not be continuous. This also means that boundedness is no longer equivalent to Lipschitz continuity in this context.

If the domain is a bornological space (for example, a pseudometrizable TVS, a Fréchet space, a normed space) then a linear operators into any other locally convex spaces is bounded if and only if it is continuous. For LF spaces, a weaker converse holds; any bounded linear map from an LF space is sequentially continuous.

If is a linear operator between two topological vector spaces and if there exists a neighborhood of the origin in such that is a bounded subset of then is continuous.^{[2]}
This fact is often summarized by saying that a linear operator that is bounded on some neighborhood of the origin is necessarily continuous.
In particular, any linear functional that is bounded on some neighborhood of the origin is continuous (even if its domain is not a normed space).

Bornological spaces are exactly those locally convex spaces for which every bounded linear operator into another locally convex space is necessarily continuous.
That is, a locally convex TVS is a bornological space if and only if for every locally convex TVS a linear operator is continuous if and only if it is bounded.^{[3]}

Every normed space is bornological.

Let be a linear operator between topological vector spaces (not necessarily Hausdorff). The following are equivalent:

- is (locally) bounded;
^{[3]} - (Definition): maps bounded subsets of its domain to bounded subsets of its codomain;
^{[3]} - maps bounded subsets of its domain to bounded subsets of its image ;
^{[3]} - maps every null sequence to a bounded sequence;
^{[3]}- A
*null sequence*is by definition a sequence that converges to the origin. - Thus any linear map that is sequentially continuous at the origin is necessarily a bounded linear map.

- A
- maps every Mackey convergent null sequence to a bounded subset of
^{[note 1]}- A sequence is said to be
*Mackey convergent to the origin in*if there exists a divergent sequence of positive real number such that is a bounded subset of

- A sequence is said to be

if and are locally convex then the following may be add to this list:

- maps bounded disks into bounded disks.
^{[4]} - maps bornivorous disks in into bornivorous disks in
^{[4]}

if is a bornological space and is locally convex then the following may be added to this list:

- is sequentially continuous at some (or equivalently, at every) point of its domain.
^{[5]}- A sequentially continuous linear map between two TVSs is always bounded,
^{[1]}but the converse requires additional assumptions to hold (such as the domain being bornological and the codomain being locally convex). - If the domain is also a sequential space, then is sequentially continuous if and only if it is continuous.

- A sequentially continuous linear map between two TVSs is always bounded,
- is sequentially continuous at the origin.

- Any linear operator between two finite-dimensional normed spaces is bounded, and such an operator may be viewed as multiplication by some fixed matrix.
- Any linear operator defined on a finite-dimensional normed space is bounded.
- On the sequence space of eventually zero sequences of real numbers, considered with the norm, the linear operator to the real numbers which returns the sum of a sequence is bounded, with operator norm 1. If the same space is considered with the norm, the same operator is not bounded.
- Many integral transforms are bounded linear operators. For instance, if is a continuous function, then the operator defined on the space of continuous functions on endowed with the uniform norm and with values in the space with given by the formula is bounded. This operator is in fact a compact operator. The compact operators form an important class of bounded operators.
- The Laplace operator (its domain is a Sobolev space and it takes values in a space of square-integrable functions) is bounded.
- The shift operator on the Lp space of all sequences of real numbers with is bounded. Its operator norm is easily seen to be

Let be the space of all trigonometric polynomials on with the norm

The operator that maps a polynomial to its derivative is not bounded. Indeed, for with we have while as so is not bounded.

The space of all bounded linear operators from to is denoted by .

- is a normed vector space.
- If is Banach, then so is ; in particular, dual spaces are Banach.
- For any the kernel of is a closed linear subspace of .
- If is Banach and is nontrivial, then is Banach.

- Bounded set (topological vector space) – Generalization of boundedness
- Contraction (operator theory) – Bounded operators with sub-unit norm
- Discontinuous linear map
- Continuous linear operator
- Local boundedness
- Norm (mathematics) – Length in a vector space
- Operator algebra – Branch of functional analysis
- Operator norm – Measure of the "size" of linear operators
- Operator theory – Mathematical field of study
- Seminorm – nonnegative-real-valued function on a real or complex vector space that satisfies the triangle inequality and is absolutely homogenous
- Unbounded operator – Linear operator defined on a dense linear subspace

**^**Proof: Assume for the sake of contradiction that converges to but is not bounded in Pick an open balanced neighborhood of the origin in such that does not absorb the sequence Replacing with a subsequence if necessary, it may be assumed without loss of generality that for every positive integer The sequence is Mackey convergent to the origin (since is bounded in ) so by assumption, is bounded in So pick a real such that for every integer If is an integer then since is balanced, which is a contradiction. Q.E.D. This proof readily generalizes to give even stronger characterizations of " is bounded." For example, the word "such that is a bounded subset of " in the definition of "Mackey convergent to the origin" can be replaced with "such that in "

- ^
^{a}^{b}Wilansky 2013, pp. 47–50. **^**Narici & Beckenstein 2011, pp. 156–175.- ^
^{a}^{b}^{c}^{d}^{e}Narici & Beckenstein 2011, pp. 441–457. - ^
^{a}^{b}Narici & Beckenstein 2011, p. 444. **^**Narici & Beckenstein 2011, pp. 451–457.

- "Bounded operator",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - Kreyszig, Erwin:
*Introductory Functional Analysis with Applications*, Wiley, 1989 - Narici, Lawrence; Beckenstein, Edward (2011).
*Topological Vector Spaces*. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. - Wilansky, Albert (2013).
*Modern Methods in Topological Vector Spaces*. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.