Bounded operator

Summary

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).

In normed vector spaces edit

Every bounded operator is Lipschitz continuous at  

Equivalence of boundedness and continuity edit

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

Proof

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.

In topological vector spaces edit

A linear operator   between two topological vector spaces (TVSs) is called a bounded linear operator 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.

Continuity and boundedness edit

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 edit

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.

Characterizations of bounded linear operators edit

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

  1.   is (locally) bounded;[3]
  2. (Definition):   maps bounded subsets of its domain to bounded subsets of its codomain;[3]
  3.   maps bounded subsets of its domain to bounded subsets of its image  ;[3]
  4.   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.
  5.   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  

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

  1.   maps bounded disks into bounded disks.[4]
  2.   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:

  1.   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.
  2.   is sequentially continuous at the origin.

Examples edit

  • 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  

Unbounded linear operators edit

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.

Properties of the space of bounded linear operators edit

  • 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.

See also edit

References edit

  1. ^ 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  "
  1. ^ a b Wilansky 2013, pp. 47–50.
  2. ^ Narici & Beckenstein 2011, pp. 156–175.
  3. ^ a b c d e Narici & Beckenstein 2011, pp. 441–457.
  4. ^ a b Narici & Beckenstein 2011, p. 444.
  5. ^ Narici & Beckenstein 2011, pp. 451–457.

Bibliography edit

  • "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.