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.

An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator.

Continuous linear operators

Characterizations of continuity

is a linear operator between two topological vector spaces (TVSs). The following are equivalent: is continuous. is continuous at some point

is locally convex then this list may be extended to include: for every continuous seminorm

are both Hausdorff locally convex spaces then this list may be extended to include: is weakly continuous and its transpose maps equicontinuous

is a sequential space (such as a pseudometrizable space) then this list may be extended to include: is sequentially continuous at some (or equivalently, at every) point of its domain.

is pseudometrizable or metrizable (such as a normed or Banach space) then we may add to this list: is a bounded linear operator).

is seminormable space (such as a normed space) then this list may be extended to include:

are both normed or seminormed spaces) then this list may be extended to include:

finite-dimensional then this list may be extended to include:

Continuity and boundedness

is a linear map between topological vector spaces (TVSs).

Bounded subset

The notion of a "bounded set" for a topological vector space is that of being a von Neumann bounded set. If the space happens to also be a normed space (or a seminormed space is von Neumann bounded if and only if it is A subset of a normed (or seminormed) space is called ) with the absolute value is contained in some open (or closed) ball centered at the origin (zero).

Any translation, scalar multiple, and subset of a bounded set is again bounded.

Function bounded on a set

is said to be is a bounded subset

Bounded linear maps

between TVSss is said to be and is called a if for every (von Neumann) bounded subset is also a (semi)normed space then this happens if and only if the operator norm is finite. Every sequentially continuous linear operator is bounded.

Function bounded on a neighborhood and local boundedness

is said to be or if there exists a neighborhood is a bounded subset It is "" (of some point) if there exists is necessarily locally bounded at point of its domain. The term "locally bounded" is sometimes used to refer to a map that is locally bounded at every point of its domain, but some functional analysis authors define "locally bounded" to instead be a synonym of "bounded linear operator", which are related but equivalent concepts. For this reason, this article will avoid the term "locally bounded" and instead say "locally bounded at every point" (there is no disagreement about the definition of "locally bounded ").

Bounded on a neighborhood implies continuous implies bounded

A linear map is "bounded on a neighborhood" (of some point) if and only if it is locally bounded at every point of its domain, in which case it is necessarily continuous (even if its domain is not a normed space) and thus also bounded (because a continuous linear operator is always a bounded linear operator).

For any linear map, if it is bounded on a neighborhood then it is continuous, and if it is continuous then it is bounded. The converse statements are not true in general but they are both true when the linear map's domain is a normed space. Examples and additional details are now given below.

Continuous and bounded but not bounded on a neighborhood

The next example shows that it is possible for a linear map to be continuous (and thus also bounded) but not bounded on any neighborhood. In particular, it demonstrates that being "bounded on a neighborhood" is always synonymous with being "bounded".

is the identity map on some locally convex topological vector space then this linear map is always continuous (indeed, even a TVS-isomorphism) and bounded which is equivalent to being a seminormable space is Hausdorff, is the same as being a normable space). This shows that it is possible for a linear map to be continuous but bounded on any neighborhood. Indeed, this example shows that every locally convex space that is not seminormable has a linear TVS-automorphism that is not bounded on any neighborhood of any point. Thus although every linear map that is bounded on a neighborhood is necessarily continuous, the converse is not guaranteed in general.

Guaranteeing converses

To summarize the discussion below, for a linear map on a normed (or seminormed) space, being continuous, being bounded, and being bounded on a neighborhood are all equivalent. A linear map whose domain codomain is normable (or seminormable) is continuous if and only if it bounded on a neighborhood. And a bounded linear operator valued in a locally convex space will be continuous if its domain is (pseudo)metrizable or bornological.

Guaranteeing that "continuous" implies "bounded on a neighborhood"

A TVS is said to be if there exists a neighborhood that is also a bounded set. For example, every normed or seminormed space). Consequently, a linear map from a locally bounded TVS into any other TVS is continuous if and only if it is bounded on a neighborhood must be a locally bounded TVS (because the identity function is always a continuous linear map).

Any linear map from a TVS into a locally bounded TVS (such as any linear functional) is continuous if and only if it is bounded on a neighborhood. is necessarily bounded on a neighborhood must be a locally bounded TVS. In particular, a linear functional on an arbitrary TVS is continuous if and only if it is bounded on a neighborhood.

Thus when the domain the codomain of a linear map is normable or seminormable, then continuity will be equivalent to being bounded on a neighborhood.

Guaranteeing that "bounded" implies "continuous"

A continuous linear operator is always a bounded linear operator. But importantly, in the most general setting of a linear operator between arbitrary topological vector spaces, it is possible for a linear operator to be bounded but to be continuous.

A linear map whose domain is pseudometrizable (such as any normed space) is bounded if and only if it is continuous. The same is true of a linear map from a bornological space into a locally convex space.

Guaranteeing that "bounded" implies "bounded on a neighborhood"

is a bounded linear operator from a normed space of the origin, which (as mentioned above) guarantees continuity.

Continuous linear functionals

Every linear functional on a topological vector space (TVS) is a linear operator so all of the properties described above for continuous linear operators apply to them. However, because of their specialized nature, we can say even more about continuous linear functionals than we can about more general continuous linear operators.

Characterizing continuous linear functionals

be a topological vector space need not be Hausdorff or locally convex be a linear functional The following are equivalent:

is continuous. is continuous at some point is continuous at the origin.

• there exists some neighborhood

  be a closed ball in this supremum is a sufficient but is instead a necessary but ). This is one of several reasons why many definitions involving linear functionals, such as polar sets) inequalities.

is bounded on a neighborhood is a locally bounded at some point of its domain.

• is a bounded subset

• Importantly, a linear functional being "bounded on a neighborhood" is in general equivalent to being a "bounded linear functional" because (as described above) it is possible for a linear map to be bounded but continuous. However, continuity and boundedness are equivalent if the domain is a normed or seminormed space; that is, for a linear functional on a normed space, being "bounded" is equivalent to being "bounded on a neighborhood".

is bounded on a neighborhood of the origin is a locally bounded at the origin.

• will satisfy the definition of continuity at the origin given in (4) above.

• which is called the (absolute) polar Polar sets, and so also this particular inequality, play important roles in duality theory.

is a locally bounded at every point of its domain. is

• is a continuous.

is closed. denotes the real part

are complex vector spaces then this list may be extended to include: is continuous.

is a sequential space then this list may be extended to include: is sequentially continuous at some (or equivalently, at every) point of its domain.

is metrizable or pseudometrizable (for example, a Fréchet space or a normed space) then this list may be extended to include: is a bounded linear operator (that is, it maps bounded subsets of its domain to bounded subsets of its codomain).

is a bornological space (for example, a pseudometrizable TVS is locally convex then this list may be extended to include: is a bounded linear operator. is sequentially continuous at some (or equivalently, at every) point of its domain. is sequentially continuous at the origin.

is a vector space over the real numbers is real-valued) then this list may be extended to include: is closed. is closed.

are continuous (respectively, bounded), or else all three are discontinuous (respectively, unbounded).

Examples

Every linear map whose domain is a finite-dimensional Hausdorff topological vector space (TVS) is continuous. This is not true if the finite-dimensional TVS is not Hausdorff.

of the origin. In particular, every TVS has a non-empty continuous dual space (although it is possible for the constant zero map to be its only continuous linear functional).

is any Hausdorff TVS. Then linear functional is closed. is necessarily a bounded linear functional if and only if every bounded subset is contained in a finite-dimensional vector subspace.

Properties

A locally convex metrizable topological vector space is normable if and only if every bounded linear functional on it is continuous.

A continuous linear operator maps bounded setss into bounded sets.

which is true due to the additivity

Properties of continuous linear functionals

is a complex normed space (where in particular, one side is infinite if and only if the other side is infinite).

is an open map.

centered at the origin then the following are equivalent:

See also

References