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
See also: Bounded operator

Suppose that F : X → Y is a linear operator between two topological vector spaces (TVSs). The following are equivalent:

  1. F is continuous at 0 in X.
  2. F is continuous at some point x0 ∈ X.
  3. F is continuous everywhere in X

and if Y is locally convex then we may add to this list:

  1. for every continuous seminorm q on Y, there exists a continuous seminorm p on X such that q ∘ F ≤ p.[1]

and if X and Y are both Hausdorff locally convex spaces then we may add to this list:

  1. F is weakly continuous and its transpose tF : Y' → X' maps equicontinuous subsets of Y' to equicontinuous subsets of X'.

and if X is pseudometrizable (i.e. if it has a countable neighborhood basis at the origin) then we may add to this list:

  1. F is a Bounded linear operator (i.e. it maps bounded subsets of X to bounded subsets of Y).[2]

and if X and Y are seminormed spaces then we may add to this list:

  1. for every ε > 0 there exists a δ > 0 such that ||x - y|| < δ implies ||Fx - Fy|| < ε;

and if Y is locally bounded then we may add to this list:

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

and if X and Y are Hausdorff locally convex TVSs with Y finite-dimensional then we may add to this list:

  1. the graph of F is closed in X × Y.[4]

Sufficient conditions for continuity

Suppose that F : X → Y is a linear operator between two TVSs.

  • If there exists a neighborhood U of 0 in X such that F(U) is a bounded subset of Y, then F is continuous.[2]
  • If X is a pseudometrizable TVS and F maps bounded subsets of X to bounded subsets of Y, then F is continuous.[2]

Properties of continuous linear operators

A locally convex metrizable TVS 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

F −1(D) + x0 = F −1(D + F(x0))}}

for any subset D of Y and any x0 ∈ X, which is true due to the additivity of F.

Continuous linear functionals

Every linear functional on a 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 X be a topological vector space (TVS) (we do not assume that X is Hausdorff or locally convex) and let f be a linear functional on X. The following are equivalent:[1]

  1. f is continuous.
  2. f is continuous at the origin.
  3. f is continuous at some point of X.
  4. f is uniformly continuous on X.
  5. There exists some neighborhood U of the origin such that f(U) is bounded.[2]
  6. The kernel of f is closed in X.[2]
  7. Either f = 0 or else the kernel of f is not dense in X.[2]
  8. Re f is continuous, where Re f denotes the real part of f.
  9. There exists a continuous seminorm p on X such that |f| ≤ p.
  10. The graph of f is closed.[5]

and if X is pseudometrizable (i.e. if it has a countable neighborhood basis at the origin) then we may add to this list:

  1. f is locally bounded (i.e. it maps bounded subsets to bounded subsets).[2]

and if in addition X is a vector space over the real numbers (which in particular, implies that f is real-valued), then we may add to this list:

  1. There exists a continuous seminorm p on X such that f ≤ p.[1]
  2. For some real r, the half-space { x ∈ X : f(x) ≤ r} is closed.
  3. The above statement but with the word "some" replaced by "any."[6]

and if X is a complex topological vector space (TVS), then we may add to this list:

  1. The imaginary part of f is continuous.

Thus, if X is a complex then either all three of f, Re f, and Im f 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 is continuous.
  • If X is a TVS, then every bounded linear functional on X is continuous if and only if every bounded subset of X is contained in a finite-dimensional vector subspace.[7]

Properties of continuous linear functionals

If X is a complex normed space and f is a linear functional on X, then ||f|| = ||Re f||[8] (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 X is an open map.[1] Note that if X is a real vector space, f is a linear functional on X, and p is a seminorm on X, then |f| ≤ p if and only if f ≤ p.[1]

See also

References
  1. Narici & Beckenstein 2011, pp. 126-128.
  2. Narici & Beckenstein 2011, pp. 156-175.
  3. Wilansky 2013, p. 54.
  4. Narici & Beckenstein 2011, p. 476.
  5. Wilansky 2013, p. 63.
  6. Narici & Beckenstein 2011, pp. 225-273.
  7. Wilansky 2013, p. 50.
  8. Narici & Beckenstein 2011, p. 128.
  • Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003.
  • Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. 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. 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. 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.
  • Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.
  • Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.
  • Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
  • Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
  • Rudin, Walter (January 1991). Functional analysis. McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
  • Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067.
  • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
  • Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.
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