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.
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

Suppose that is a linear operator between two topological vector spaces.
The following are equivalent:

  1. is continuous.
  2. is continuous at some point
  3. is continuous at the origin in
If is locally convex then this list may be extended to include:

  1. for every continuous seminorm on there exists a continuous seminorm on such that
If and are both Hausdorff locally convex spaces then this list may be extended to include:

  1. is weakly continuous and its transpose maps equicontinuous subsets of to equicontinuous subsets of
If is a sequential space then this list may be extended to include:

  1. is sequentially continuous at some point of its domain.
If is pseudometrizable or metrizable then we may add to this list:

  1. is a bounded linear operator.
If is seminormable space then this list may be extended to include:

  1. maps some neighborhood of 0 to a bounded subset of
If and are both normed or seminormed spaces then this list may be extended to include:

  1. for every there exists some such that
If and are Hausdorff locally convex spaces with finite-dimensional then this list may be extended to include:

  1. the graph of is closed in

Continuity and boundedness

Throughout, is a linear map between topological vector spaces.
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 then a subset is von Neumann bounded if and only if it is, meaning that
A subset of a normed space is called if it is norm-bounded.
For example, the scalar field with the absolute value is a normed space, so a subset is bounded if and only if is finite, which happens if and only if is contained in some open ball centered at the origin.
Any translation, scalar multiple, and subset of a bounded set is again bounded.
Function bounded on a set
If is a set then is said to be if is a bounded subset of which if is a normed space happens if and only if
A linear map is bounded on a set if and only if it is bounded on for every if and only if it is bounded on for every non-zero scalar .
Consequently, if is a normed or seminormed space, then a linear map is bounded on some non-degenerate open or closed ball if and only if it is bounded on the closed unit ball centered at the origin
Bounded linear maps
By definition, a linear map between TVSs is said to be and is called a if for every bounded subset of its domain, is a bounded subset of it codomain; or said more briefly, if it is bounded on every bounded subset of its domain. When the domain is a normed space then it suffices to check this condition for the open or closed unit ball centered at the origin. Explicitly, if denotes this ball then is a bounded linear operator if and only if is a bounded subset of if is also a 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
In contrast, a map is said to be a point or if there exists a neighborhood of this point in such that is a bounded subset of
It is "" if there exists point in its domain at which it is locally bounded, in which case this linear map is necessarily locally bounded at point of its domain.
The term "Locally bounded function|" 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".

Bounded on a neighborhood implies continuous implies bounded

A linear map is "bounded on a neighborhood" if and only if it is locally bounded at every point of its domain, in which case it is necessarily continuous and thus also bounded.
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 but not bounded on any neighborhood. In particular, it demonstrates that being "bounded on a neighborhood" is always synonymous with being "bounded".
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 space, being continuous, being bounded, and being bounded on a neighborhood are all equivalent.
A linear map whose domain codomain is normable 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 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 is a locally bounded TVS since the unit ball centered at the origin is a bounded neighborhood of the origin.
If is a bounded neighborhood of the origin in a TVS then its image under any continuous linear map will be a bounded set.
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.
Moreover, any TVS with this property must be a locally bounded TVS. Explicitly, if is a TVS such that every continuous linear map whose domain is is necessarily bounded on a neighborhood, then must be a locally bounded TVS.
Any linear map from a TVS into a locally bounded TVS is continuous if and only if it is bounded on a neighborhood.
Conversely, if is a TVS such that every continuous linear map with codomain is necessarily bounded on a neighborhood, then 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 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"
In general, without additional information about either the linear map or its domain or codomain, the map being "bounded" is not equivalent to it being "bounded on a neighborhood".
If is a bounded linear operator from a normed space into some TVS then is necessarily continuous; this is because any open ball centered at the origin in is both a bounded subset and a neighborhood of the origin in so that is thus bounded on this neighborhood of the origin, which guarantees continuity.

Continuous linear functionals

Every linear functional on a topological vector space 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

Let be a topological vector space over the field and let be a linear functional on
The following are equivalent:

  1. is continuous.
  2. is uniformly continuous on
  3. is continuous at some point of
  4. is continuous at the origin.
    • By definition, said to be continuous at the origin if for every open ball of radius centered at in the codomain there exists some neighborhood of the origin in such that
    • If is a closed ball then the condition holds if and only if
    • * It is important that be a closed ball in this supremum characterization. Assuming that is instead an open ball, then is a sufficient but condition for to be true, whereas the non-strict inequality is instead a necessary but condition for to be true. This is one of several reasons why many definitions involving linear functionals, such as polar sets for example, involve closed neighborhoods and non-strict inequalities.
  5. is bounded on a neighborhood. Said differently, is a [|locally bounded at some point] of its domain.
    • Explicitly, this means that there exists some neighborhood of some point such that is a bounded subset of that is, such that This supremum over the neighborhood is equal to if and only if
    • Importantly, a linear functional being "bounded on a neighborhood" is in general equivalent to being a "bounded linear functional" because 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".
  6. is [|bounded on a neighborhood of the origin]. Said differently, is a locally bounded at the origin.
    • The equality holds for all scalars and when then will be neighborhood of the origin. So in particular, if is a positive real number then for every positive real the set is a neighborhood of the origin and Using proves the next statement when
  7. There exists some neighborhood of the origin such that
    • This inequality holds if and only if for every real which shows that the positive scalar multiples of this single neighborhood will satisfy the definition of continuity at the origin given in above.
    • By definition of the set which is called the polar of the inequality holds if and only if Polar sets, and so also this particular inequality, play important roles in duality theory.
  8. is a locally bounded at every point of its domain.
  9. The kernel of is closed in
  10. Either or else the kernel of is dense in
  11. There exists a continuous seminorm on such that
    • In particular, is continuous if and only if the seminorm is a continuous.
  12. The graph of is closed.
  13. is continuous, where denotes the real part of
If and are complex vector spaces then this list may be extended to include:

  1. The imaginary part of is continuous.
If the domain is a sequential space then this list may be extended to include:

  1. is sequentially continuous at some point of its domain.
If the domain is metrizable or pseudometrizable then this list may be extended to include:

  1. is a bounded linear operator.
If the domain is a bornological space and is locally convex then this list may be extended to include:

  1. is a bounded linear operator.
  2. is sequentially continuous at some point of its domain.
  3. is sequentially continuous at the origin.
and if in addition is a vector space over the real numbers then this list may be extended to include:

  1. There exists a continuous seminorm on such that
  2. For some real the half-space is closed.
  3. For any real the half-space is closed.
If is complex then either all three of and are continuous, or else all three are discontinuous.