In functional analysis and operator theory, a bounded linear operator is a special kind of linear transformation that is particularly important in infinite dimensions. In finite dimensions, a linear transformation takes a bounded set to another bounded set (for example, a rectangle in the plane goes either to a parallelogram or bounded line segment when a linear transformation is applied). However, in infinite dimensions, linearity is not enough to ensure that bounded sets remain bounded: a bounded linear operator is thus a linear transformation that sends bounded sets to bounded sets.

between topological vector spaces that maps bounded are normed vector spaces is called the operator norm A linear operator between normed spaces is continuous if and only if it is bounded.

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

is called "bounded" then this usually means that its image 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).

In normed vector spaces

Every bounded operator is Lipschitz continuous

Equivalence of boundedness and continuity

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

Relative boundedness

Given two partially defined.

In Hilbert spaces

on a Hilbert space H becomes a C*-algebra and especially an operator space. It is possible to define various different notions of boundedness for an operator T.

for all natural numbers n. This condition implies that T is bounded, of course, but the converse need not be true.

. Again this condition implies power boundedness and norm boundedness, but the converse need not be true.

is called completely polynomially bounded and for all natural numbers n. Here, the respective matrix norms can be understood as the polynomial functional calculus. Every completely polynomially bounded operator is polynomially- and power bounded, as well as norm bounded, but the opposite does not hold in general.

Positive examples of completely polynomially bounded operators are contractive operators T,holds true.

In topological vector spaces

between two topological vector spaces (TVSs) is called a or just is bounded A subset of a TVS is called bounded (or more precisely, von Neumann bounded) if every neighborhood of the origin absorbs it. In a normed space (and even in a seminormed space), a subset is von Neumann bounded if and only if it is norm bounded. Hence, for normed spaces, the notion of a von Neumann bounded set is identical to the usual notion of a norm-bounded subset.

Continuity and boundedness

Every sequentially continuous linear operator between TVS is a bounded operator. 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.

is continuous. 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

is continuous if and only if it is bounded.

Every normed space is bornological.

Characterizations of bounded linear operators

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

1. is (locally) bounded;

2. 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;

maps every null sequence to a bounded sequence;

• 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.

• is said to be Mackey convergent to the origin

are locally convex then the following may be add to this list: maps bounded disks into bounded disks. maps bornivorous

is a bornological space 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.

• A sequentially continuous linear map between two TVSs is always bounded, but the converse requires additional assumptions to hold (such as the domain being bornological and the codomain being locally convex).

• is also a sequential space is sequentially continuous if and only if it is continuous.

is sequentially continuous at the origin.

Examples

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 norm, the same operator is not bounded. Many integral transforms endowed with the uniform norm 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 unilateral shift operator on the Lp space of all sequences

Unbounded linear operators

be the space of all trigonometric polynomials with the norm

that maps a polynomial to its derivative is not bounded.

Properties of the space of bounded linear operators

.

• is a normed vector space.

• ; in particular, dual spaces are Banach.

• .

• is Banach.

See also

References

Bibliography

• Kreyszig, Erwin: Introductory Functional Analysis with Applications, Wiley, 1989