Continuous linear operator

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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 (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 boundedness[]

Throughout, is a linear map between topological vector spaces. The notion of "bounded set" for a topological vector space (TVS) is that of being a von Neumann bounded set. If a TVS happens to also be a normed (or seminormed) space then a subset is von Neumann bounded if and only if it is norm bounded; that is, if and only if

"Bounded" versus "continuous"

By definition, a linear map between topological vector spaces (TVSs) is said to be bounded and is called a bounded linear operator if it maps (von Neumann) bounded subsets of the domain to bounded subsets of the codomain. Every sequentially continuous linear operator is bounded.[5] So in particular, a continuous linear operator is always a bounded linear operator[6] but in general, a bounded linear operator need not be continuous.

A linear map whose domain is pseudometrizable is bounded if and only if it is continuous.[2] Every bounded linear operator from a bornological space into a locally convex space is continuous.[6]

"Bounded on a neighborhood" versus "continuous"

If is a set then is said to be bounded on if is a bounded subset of and it is said to be unbounded on otherwise. In particular, 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 (in contrast, is call bounded if for every that is a bounded subset of its image is a bounded subset of ). A linear map is locally bounded at every point of its domain if and only if there exists a point in its domain at which it is locally bounded; or said differently, if and only if it is "bounded on a neighborhood" (of some point).

A map is called locally bounded if it is locally bounded at every point of its domain, but some functional analysis authors define "locally bounded" to be a synonym of "bounded linear operator" (which are related but not equivalent concepts), which is why 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").

A linear map that is bounded on a neighborhood (of some/every point) is necessarily continuous[2] (even if its domain is not a normed space). The next example shows that the converse is not always guaranteed.

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.

However, a linear map from a TVS into a normed or seminormed space (such as a linear functional for example) is continuous if and only if it is bounded on some neighborhood. In particular, a linear functional on a arbitrary TVS is continuous if and only if it is bounded on some neighborhood. In addition, a linear map from a normed or seminormed space into a TVS is continuous if and only if it is bounded on a neighborhood. Thus when the domain or codomain of a linear map is normable or seminormable, then continuity is equivalent to being bounded on a neighborhood.

"Bounded on a neighborhood" versus "bounded"

A linear map being "bounded on a neighborhood" is not the same as it being "bounded". Because a continuous linear operator is always a bounded linear operator,[6] if a linear operator is "bounded on a neighborhood" then it is necessarily (continuous and thus also) bounded. 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.

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" (meaning that it is a bounded linear operator) but to not be continuous. When the domain is metrizable or bornological, such as when it is a normed space, and the codomain is locally convex, then a linear operator being "bounded" is equivalent to it being continuous.[6] But without additional information about either the linear map or it's domain or codomain, the map being "bounded" is not equivalent to it being "bounded on a neighborhood".

In summary, 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.

Properties of continuous linear operators[]

A locally convex metrizable topological vector space is normable if and only if every 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

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[]

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
    • 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 by replacing with an appropriate choice of where the value of can also be scaled as desired (if it is not ).
  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.
    • Moreover, is bounded on a set if and only if is bounded on for every (because ).
  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.[7]
  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.[8]
  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."[9]

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

Sufficient conditions for continuous linear functionals[]

  • Every linear function on a finite-dimensional Hausdorff topological vector space (TVS) is continuous. This is not true if the finite-dimensional TVS is not Hausdorff.
  • If is a TVS, then every bounded linear functional on is continuous if and only if every bounded subset of is contained in a finite-dimensional vector subspace.[10]

Properties of continuous linear functionals[]

If is a complex normed space and is a linear functional on then [11] (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 also[]

References[]

  1. ^ a b c d e Narici & Beckenstein 2011, pp. 126–128.
  2. ^ a b c d e f g h i 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, p. 63.
  8. ^ Narici & Beckenstein 2011, pp. 451–457.
  9. ^ Narici & Beckenstein 2011, pp. 225–273.
  10. ^ Wilansky 2013, p. 50.
  11. ^ Narici & Beckenstein 2011, p. 128.
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  • Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. Vol. 15. New York: Springer. ISBN 978-0-387-90081-0. OCLC 878109401.
  • Bourbaki, Nicolas (1987) [1981]. Sur certains espaces vectoriels topologiques [Topological Vector Spaces: Chapters 1–5]. Annales de l'Institut Fourier. Éléments de mathématique. Vol. 2. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190.
  • Conway, John (1990). A course in functional analysis. Graduate Texts in Mathematics. Vol. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.
  • Dunford, Nelson (1988). Linear operators (in Romanian). New York: Interscience Publishers. ISBN 0-471-60848-3. OCLC 18412261.
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  • Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
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