In functional analysis, 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 in

- In topological vector spaces
- Bornological spaces
- Characterizations of bounded linear operators
- Bounded linear operators between normed spaces
- Further properties
- Examples
- Unbounded linear operators
- Properties of the space of bounded linear operators
- See also
- References
- Bibliography

The smallest such denoted by is called the operator norm of

A linear operator that is sequentially continuous or continuous is a bounded operator and moreover, a linear operator between normed spaces is bounded if and only if it is continuous. However, a bounded linear operator between more general topological vector spaces is not necessarily continuous.

A linear operator between two topological vector spaces (TVSs) is **locally bounded** or just **bounded** if whenever is bounded in then is bounded in 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.

Every sequentially continuous linear operator between TVS is a bounded operator.^{ [1] } This implies that every continuous linear operator 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. Clearly, this also means that boundedness is no longer equivalent to Lipschitz continuity in this context.

If the domain is a bornological space (e.g. 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.

Bornological spaces are exactly those locally convex spaces for 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.^{ [2] }

Every normed space is bornological.

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

- is (locally) bounded;
^{ [2] } - (Definition): maps bounded subsets of its domain to bounded subsets of its codomain;
^{ [2] } - maps bounded subsets of its domain to bounded subsets of its image ;
^{ [2] } - maps every null sequence to a bounded sequence;
^{ [2] }- 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

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

- maps bounded disks into bounded disks.
^{ [3] } - maps bornivorous disks in into bornivorous disks in
^{ [3] }

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

- is sequentially continuous.
^{ [4] } - is sequentially continuous at the origin.

A bounded linear operator is generally not a bounded function, as generally one can find a sequence in such that Instead, all that is required for the operator to be bounded is that

for all So, the operator could only be a bounded function if it satisfied for all as is easy to understand by considering that for a linear operator, for all scalars Rather, a bounded linear operator is a locally bounded function.

A linear operator between normed spaces is bounded if and only if it is continuous, and by linearity, if and only if it is continuous at zero.

- Equivalence of boundedness and continuity

A linear operator between normed spaces and is bounded if and only if it is a continuous linear operator. A proof is given below.

Proof of equivalence of boundedness and continuity |
---|

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. |

The condition for to be bounded, namely that there exists some such that for all

is precisely the condition for to be Lipschitz continuous at 0 (and hence everywhere, because is linear).

A common procedure for defining a bounded linear operator between two given Banach spaces is as follows. First, define a linear operator on a dense subset of its domain, such that it is locally bounded. Then, extend the operator by continuity to a continuous linear operator on the whole domain.

- 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 compact. 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

Not every linear operator between normed spaces is bounded. Let be the space of all trigonometric polynomials defined on with the norm

Define the operator which acts by taking the derivative, so it maps a polynomial to its derivative Then, for

with we have while as so this operator is not bounded.

It turns out that this is not a singular example, but rather part of a general rule. However, given any normed spaces and with infinite-dimensional and not being the zero space, one can find a linear operator which is not continuous from to

That such a basic operator as the derivative (and others) is not bounded makes it harder to study. If, however, one defines carefully the domain and range of the derivative operator, one may show that it is a closed operator. Closed operators are more general than bounded operators but still "well-behaved" in many ways.

- The space of all bounded linear operators from to is denoted by and is a normed vector space.
- If is Banach, then so is
- from which it follows that 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)
- Discontinuous linear map
- Continuous linear operator
- Norm (mathematics) – Length in a vector space
- Normed space
- Operator algebra – Branch of functional analysis
- Operator norm – Measure of the "size" of linear operators
- Operator theory
- Seminorm
- Unbounded operator
- Topological vector space – Vector space with a notion of nearness

In mathematics, more specifically in functional analysis, a **Banach space** is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

In mathematics, any vector space * has a corresponding ***dual vector space** consisting of all linear forms on *, together with the vector space structure of pointwise addition and scalar multiplication by constants.*

In mathematics, a **topological vector space** is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

In the area of mathematics known as functional analysis, a **reflexive space** is a locally convex topological vector space (TVS) such that the canonical evaluation map from into its bidual is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space is reflexive if and only if the canonical evaluation map from into its bidual is surjective; in this case the normed space is necessarily also a Banach space. In 1951, R. C. James discovered a Banach space, now known as James' space, that is *not* reflexive but is nevertheless isometrically isomorphic to its bidual.

In functional analysis and related areas of mathematics, **Fréchet spaces**, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically *not* Banach spaces.

In mathematics, the **uniform boundedness principle** or **Banach–Steinhaus theorem** is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm.

In functional analysis and related areas of mathematics, **locally convex topological vector spaces** (**LCTVS**) or **locally convex spaces** are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

In functional analysis and related branches of mathematics, the **Banach–Alaoglu theorem** states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak-* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.

In functional analysis and related areas of mathematics, a **sequence space** is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field *K* of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in *K*, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.

In functional analysis and related areas of mathematics, a **barrelled space** is a topological vector spaces (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A **barrelled set** or a **barrel** in a topological vector space is a set that is convex, balanced, absorbing, and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them.

In functional analysis and related areas of mathematics, a set in a topological vector space is called **bounded** or **von Neumann bounded**, if every neighborhood of the zero vector can be *inflated* to include the set. A set that is not bounded is called **unbounded**.

In mathematics, particularly in functional analysis, a **bornological space** is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by that property that a linear map from a bornological space into any locally convex spaces is continuous if and only if it is a bounded linear operator.

In mathematics, a **nuclear space** is a topological vector space with many of the good properties of finite-dimensional vector spaces. The topology on them can be defined by a family of seminorms whose unit balls decrease rapidly in size. Vector spaces whose elements are "smooth" in some sense tend to be nuclear spaces; a typical example of a nuclear space is the set of smooth functions on a compact manifold.

In mathematics, particularly in functional analysis, a **webbed space** is a topological vector space designed with the goal of allowing the results of the open mapping theorem and the closed graph theorem to hold for a wider class of linear maps whose codomains are webbed spaces. A space is called webbed if there exists a collection of sets, called a *web* that satisfies certain properties. Webs were first investigated by de Wilde.

In functional analysis and related areas of mathematics, a **complete topological vector space** is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point towards which they all get closer to. The notion of "points that get progressively closer" is made rigorous by *Cauchy nets* or *Cauchy filters*, which are generalizations of *Cauchy sequences*, while "point towards which they all get closer to" means that this net or filter converges to Unlike the notion of completeness for metric spaces, which it generalizes, the notion of completeness for TVSs does not depend on any metric and is defined for *all* TVSs, including those that are not metrizable or Hausdorff.

In functional analysis, two methods of constructing normed spaces from disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces. One method is used if the disk is bounded: in this case, the **auxiliary normed space** is with norm The other method is used if the disk is absorbing: in this case, the auxiliary normed space is the quotient space If the disk is both bounded and absorbing then the two auxiliary normed spaces are canonically isomorphic.

In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spaces (TVSs).

This is a glossary for the terminology in a mathematical field of functional analysis.

In functional analysis and related areas of mathematics, a **metrizable** topological vector space (TVS) is a TVS whose topology is induced by a metric. An **LM-space** is an inductive limit of a sequence of locally convex metrizable TVS.

In mathematics, particularly in functional analysis and topology, the **closed graph theorem** is a fundamental result stating that a linear operator with a closed graph will, under certain conditions, be continuous. The original result has been generalized many times so there are now many theorems referred to as "closed graph theorems."

- ↑ 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. 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 "

- ↑ Wilansky 2013, pp. 47-50.
- 1 2 3 4 5 Narici & Beckenstein 2011, pp. 441-457.
- 1 2 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.

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