Bounded operator
Encyclopedia
In functional analysis
, a branch of mathematics
, a bounded linear operator is a linear transformation
L between normed vector space
s X and Y for which the ratio of the norm of L(v) to that of v is bounded
by the same number, over all non-zero vectors v in X. In other words, there exists some M > 0 such that for all v in X
The smallest such M is called the operator norm
of L.
A bounded linear operator is generally not a bounded function
; the latter would require that the norm of L(v) be bounded for all v, which is not possible unless Y is the zero vector space. Rather, a bounded linear operator is a locally bounded function.
A linear operator is bounded if and only if it is continuous
.
. The proof is as follows.
s defined on [−π, π], with the norm
Define the operator L:X→X which acts by taking the derivative
, so it maps a polynomial P to its derivative P′. Then, for
with n=1, 2, ...., we have
while 
as n→∞, so this operator is not bounded.
It turns out that this is not a singular example, but rather part of a general rule. Any linear operator defined on a finite-dimensional normed space is bounded. However, given any normed spaces X and Y with X infinite-dimensional and Y not being the zero space, one can find a linear operator which is not continuous
from X to Y.
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.
is precisely the condition for L to be Lipschitz continuous at 0 (and hence, everywhere, because L 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 (see continuous linear extension).
to a bounded set, and here is meant the more general condition of boundedness for sets in a topological vector space
(TVS): a set is bounded if and only if it is absorbed by every neighborhood of 0. Note that the two notions of boundedness coincide for locally convex spaces.
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.
A converse does hold when the domain is pseudometrisable, a case which includes Fréchet space
s. For LF spaces, a weaker converse holds; any bounded linear map from an LF space is sequentially continuous.
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a bounded linear operator is a linear transformation
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
L between normed vector space
Normed vector space
In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....
s X and Y for which the ratio of the norm of L(v) to that of v is bounded
Bounded set
In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded...
by the same number, over all non-zero vectors v in X. In other words, there exists some M > 0 such that for all v in X
The smallest such M is called the operator norm
Operator norm
In mathematics, the operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.- Introduction and definition :...
of L.A bounded linear operator is generally not a bounded function
Bounded function
In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M...
; the latter would require that the norm of L(v) be bounded for all v, which is not possible unless Y is the zero vector space. Rather, a bounded linear operator is a locally bounded function.
A linear operator is bounded if and only if it is continuous
Continuous linear operator
In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces....
.
Examples
- Any linear operator between two finite-dimensional normed spaces is bounded, and such an operator may be viewed as multiplication by some fixed matrixMatrix (mathematics)In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
.
- 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 compactCompact operatorIn functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y...
. The compact operators form an important class of bounded operators.
- The Laplace operatorLaplace operatorIn mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...
-
is bounded.
- The shift operatorShift operatorIn mathematics, and in particular functional analysis, the shift operator or translation operator is an operator that takes a function to its translation . In time series analysis, the shift operator is called the lag operator....
on the l2 spaceLp spaceIn mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
of all sequenceSequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
s (x0, x1, x2...) of real numbers with
-
- is bounded. Its norm is easily seen to be 1.
Equivalence of boundedness and continuity
As stated in the introduction, a linear operator L between normed spaces X and Y is bounded if and only if it is a continuous linear operatorContinuous linear operator
In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces....
. The proof is as follows.
- Suppose that L is bounded. Then, for all vectors v and h in X with h nonzero we have
-
- Letting
go to zero shows that L is continuous at v. Moreover, since the constant M does not depend on v, this shows that in fact L is uniformly continuousUniform continuityIn mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f and f be as close to each other as we please by requiring only that x and y are sufficiently close to each other; unlike ordinary continuity, the maximum distance between x and y cannot...
(Even stronger, it is Lipschitz continuous.)
- Conversely, it follows from the continuity at the zero vector that there exists a
such that 
for all vectors h in X with 
. Thus, for all non-zero 
in X, one has
-
- This proves that L is bounded.
Linearity and boundedness
Not every linear operator between normed spaces is bounded. Let X be the space of all trigonometric polynomialTrigonometric polynomial
In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin and cos with n a natural number. The coefficients may be taken as real numbers, for real-valued functions...
s defined on [−π, π], with the norm
Define the operator L:X→X which acts by taking the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
, so it maps a polynomial P to its derivative P′. Then, for
with n=1, 2, ...., we have
while as n→∞, so this operator is not bounded.It turns out that this is not a singular example, but rather part of a general rule. Any linear operator defined on a finite-dimensional normed space is bounded. However, given any normed spaces X and Y with X infinite-dimensional and Y not being the zero space, one can find a linear operator which is not continuous
Discontinuous linear map
In mathematics, linear maps form an important class of "simple" functions which preserve the algebraic structure of linear spaces and are often used as approximations to more general functions . If the spaces involved are also topological spaces , then it makes sense to ask whether all linear maps...
from X to Y.
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 operator
In mathematics, specifically in functional analysis, closed linear operators are an important class of linear operators on Banach spaces. They are more general than bounded operators, and therefore not necessarily continuous, but they still retain nice enough properties that one can define the...
. Closed operators are more general than bounded operators but still "well-behaved" in many ways.
Further properties
The condition for L to be bounded, namely that there exists some M such that for all vis precisely the condition for L to be Lipschitz continuous at 0 (and hence, everywhere, because L is linear).
A common procedure for defining a bounded linear operator between two given Banach
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
spaces is as follows. First, define a linear operator on a dense subset
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
of its domain, such that it is locally bounded. Then, extend the operator by continuity to a continuous linear operator on the whole domain (see continuous linear extension).
Properties of the space of bounded linear operators
- The space of all bounded linear operators from U to V is denoted by B(U,V) and is a normed vector space.
- If V is Banach, then so is B(U,V),
- from which it follows that dual spaceDual spaceIn mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
s are Banach. - For any A in B(U,V), the kernel of A is a closed linear subspace of U.
- If B(U,V) is Banach and U is nontrivial, then V is Banach.
Topological vector spaces
The boundedness condition for linear operators on normed spaces can be restated. An operator is bounded if it takes every bounded setBounded set (topological vector space)
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...
to a bounded set, and here is meant the more general condition of boundedness for sets in a topological vector space
Topological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
(TVS): a set is bounded if and only if it is absorbed by every neighborhood of 0. Note that the two notions of boundedness coincide for locally convex spaces.
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.
A converse does hold when the domain is pseudometrisable, a case which includes Fréchet space
Fréchet space
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...
s. For LF spaces, a weaker converse holds; any bounded linear map from an LF space is sequentially continuous.