Bounded operator

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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

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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.

In topological vector spaces

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

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.

Characterizations of bounded linear operators

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

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

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

  1. maps bounded disks into bounded disks. [3]
  2. 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:

  1. is sequentially continuous. [4]
  2. is sequentially continuous at the origin.

Bounded linear operators between normed spaces

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.

Further properties

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.

Examples

Unbounded linear operators

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.

Properties of the space of bounded linear operators

See also

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References

  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. 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 "

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