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).
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 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:
(Definition): 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;[3]
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.
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:
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).
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.
Seminorm – nonnegative-real-valued function on a real or complex vector space that satisfies the triangle inequality and is absolutely homogenousPages displaying wikidata descriptions as a fallback
^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 "
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.