Continuous linear operator

Summary

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 operatorsEdit

Characterizations of continuityEdit

Suppose that   is a linear operator between two topological vector spaces (TVSs). 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  [1]

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 (such as a pseudometrizable space) then this list may be extended to include:

  1.   is sequentially continuous at some (or equivalently, at every) point of its domain.

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

  1.   is a bounded linear operator (that is, it maps bounded subsets of   to bounded subsets of  ).[2]

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

  1.   maps some neighborhood of 0 to a bounded subset of  [3]

if   and   are both normed or seminormed spaces (with both seminorms denoted by  ) 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  [4]

Continuity and boundednessEdit

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

Bounded on a set

The notion of "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), such as the scalar field with the absolute value for instance, then a subset   is von Neumann bounded if and only if it is norm bounded; that is, if and only if   If   is a set then   is said to be bounded on   if   is a bounded subset of   which if   is a normed (or seminormed) space happens if and only if   A linear map   is bounded on a set   if and only if it is bounded on   for every   (because   and any translation of a bounded set is again bounded).

Bounded linear maps

By definition, a linear map   between TVSs is said to be bounded and is called a bounded linear operator if for every (von Neumann) 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 (or seminormed) 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 (semi)normed space then this happens if and only if the operator norm   is finite. Every sequentially continuous linear operator is bounded.[5]

Bounded on a neighborhood and local boundedness

In contrast, a map   is said to be bounded on a neighborhood of a point   or locally bounded at   if there exists a neighborhood   of this point in   such that   is a bounded subset of   It is "bounded on a neighborhood" (of some point) if there exists some point   in its domain at which it is locally bounded, in which case this linear map   is necessarily locally bounded at every 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 not 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 at a point").

Bounded on a neighborhood implies continuous implies boundedEdit

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[2] (even if its domain is not a normed space) and thus also bounded (because a continuous linear operator is always a bounded linear operator).[6]

For any linear map, if it is bounded on a neighborhood then it is continuous,[2][7] and if it is continuous then it is bounded.[6] 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 neighborhoodEdit

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 not always synonymous with being "bounded".

Example: A continuous and bounded linear map that is not bounded on any neighborhood: If   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, but   is bounded on a neighborhood if and only if there exists a bounded neighborhood of the origin in   which is equivalent to   being a seminormable space (which if   is Hausdorff, is the same as being a normable space). This shows that it is possible for a linear map to be continuous but not 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 conversesEdit

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

A TVS is said to be locally bounded if there exists a neighborhood that is also a bounded set.[8] 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 (locally bounded) TVS then its image under any continuous linear map will be a bounded set (so this map is thus bounded on this neighborhood  ). 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 (into any TVS) whose domain is   is necessarily bounded on a neighborhood, then   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.[8] Conversely, if   is a TVS such that every continuous linear map (from any TVS) with codomain   is necessarily bounded on a neighborhood, then   must be a locally bounded TVS.[8] In particular, a linear functional on a arbitrary TVS is continuous if and only if it is bounded on a neighborhood.[8]

Thus when the domain or 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.[6] 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 not be continuous.

A linear map whose domain is pseudometrizable (such as any normed space) is bounded if and only if it is continuous.[2] The same is true of a linear map from a bornological space into a locally convex space.[6]

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 (which implies that   is bounded since   is a bounded linear map) and a neighborhood of the origin in   so that   is thus bounded on this neighborhood   of the origin, which (as mentioned above) guarantees continuity.

Continuous linear functionalsEdit

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 functionalsEdit

Let   be a topological vector space (TVS) over the field   (  need not be Hausdorff or locally convex) and let   be a linear functional on   The following are equivalent:[1]

  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 (or closed) 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  
      • However, assuming that   is instead an open ball, then   is a sufficient but not necessary condition for   to be true (consider for example when   is the identity map on   and  ), whereas the non-strict inequality   is instead a necessary but not sufficient condition for   to be true (consider for example   and the closed neighborhood  ). This is one of several reasons why many definitions involving linear functionals, such as polar sets for example, involve closed (rather than open) neighborhoods and non-strict   (rather than strict ) inequalities.
  5.   is bounded on a neighborhood (of some point). 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  [2] 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 not equivalent to being a "bounded linear functional" because (as described above) it is possible for a linear map to be bounded but not 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 also a neighborhood of the origin and  
  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 (4) above.
    • By definition of the set   which is called the (absolute) polar of   the inequality   holds if and only if   Polar sets, and thus 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  [2]
  10. Either   or else the kernel of   is not dense in  [2]
  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.[9]
  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 (or equivalently, at every) point of its domain.[2]

if the domain   is metrizable or pseudometrizable (for example, a Fréchet space or a normed space) then this list may be extended to include:

  1.   is a bounded linear operator (that is, it maps bounded subsets to bounded subsets).[2]

if the domain   is a bornological space (for example, a pseudometrizable TVS) and   is locally convex then this list may be extended to include:

  1.   is a bounded linear operator.[2]
  2.   is sequentially continuous at some (or equivalently, at every) point of its domain.[10]
  3.   is sequentially continuous at the origin.

and if in addition   is a vector space over the real numbers (which in particular, implies that   is real-valued) then this list may be extended to include:

  1. There exists a continuous seminorm   on   such that  [1]
  2. For some real   the half-space   is closed.
  3. The above statement but with the word "some" replaced by "any."[11]

Thus, if   is a complex then either all three of     and   are continuous (resp. bounded), or else all three are discontinuous (resp. unbounded).

ExamplesEdit

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.

Suppose   is any Hausdorff TVS. Then every linear functional on   is necessarily continuous if and only if every vector subspace of   is closed.[12] Every linear functional on   is necessarily a bounded linear functional if and only if every bounded subset of   is contained in a finite-dimensional vector subspace.[13]

PropertiesEdit

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 sets into bounded sets.

The proof uses the facts that the translation of an open set in a linear topological space is again an open set, and the equality

 
for any subset   of   and any   which is true due to the additivity of  

Properties of continuous linear functionalsEdit

If   is a complex normed space and   is a linear functional on   then  [14] (where in particular, one side is infinite if and only if the other side is infinite).

Every non-trivial continuous linear functional on a TVS   is an open map.[1] Note that if   is a real vector space,   is a linear functional on   and   is a seminorm on   then   if and only if  [1]

If   is a linear functional and   is a non-empty subset, then by defining the sets

 
the supremum   can be written more succinctly as   because
 
If   is a scalar then
 
so that if   is a real number and   is the closed ball of radius   centered at the origin then
 

See alsoEdit

ReferencesEdit

  1. ^ a b c d e Narici & Beckenstein 2011, pp. 126–128.
  2. ^ a b c d e f g h i j Narici & Beckenstein 2011, pp. 156–175.
  3. ^ Wilansky 2013, p. 54.
  4. ^ Narici & Beckenstein 2011, p. 476.
  5. ^ Wilansky 2013, pp. 47–50.
  6. ^ a b c d Narici & Beckenstein 2011, pp. 441–457.
  7. ^ Wilansky 2013, pp. 54–55.
  8. ^ a b c d Wilansky 2013, pp. 53–55.
  9. ^ Wilansky 2013, p. 63.
  10. ^ Narici & Beckenstein 2011, pp. 451–457.
  11. ^ Narici & Beckenstein 2011, pp. 225–273.
  12. ^ Wilansky 2013, p. 55.
  13. ^ Wilansky 2013, p. 50.
  14. ^ Narici & Beckenstein 2011, p. 128.
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