Topological homomorphism

In functional analysis, a topological homomorphism or simply homomorphism (if no confusion will arise) is the analog of homomorphisms for the category of topological vector spaces (TVSs). This concept is of considerable importance in functional analysis and the famous open mapping theorem gives a sufficient condition for a continuous linear map between Fréchet spaces to be a topological homomorphism.

Definitions

edit

A topological homomorphism or simply homomorphism (if no confusion will arise) is a continuous linear map   between topological vector spaces (TVSs) such that the induced map   is an open mapping when   which is the image of   is given the subspace topology induced by  [1] This concept is of considerable importance in functional analysis and the famous open mapping theorem gives a sufficient condition for a continuous linear map between Fréchet spaces to be a topological homomorphism.

A TVS embedding or a topological monomorphism[2] is an injective topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a topological embedding.

Characterizations

edit

Suppose that   is a linear map between TVSs and note that   can be decomposed into the composition of the following canonical linear maps:

 

where   is the canonical quotient map and   is the inclusion map.

The following are equivalent:

  1.   is a topological homomorphism
  2. for every neighborhood base   of the origin in     is a neighborhood base of the origin in  [1]
  3. the induced map   is an isomorphism of TVSs[1]

If in addition the range of   is a finite-dimensional Hausdorff space then the following are equivalent:

  1.   is a topological homomorphism
  2.   is continuous[1]
  3.   is continuous at the origin[1]
  4.   is closed in  [1]

Sufficient conditions

edit

Theorem[1] — Let   be a surjective continuous linear map from an LF-space   into a TVS   If   is also an LF-space or if   is a Fréchet space then   is a topological homomorphism.

Theorem[3] — Suppose   be a continuous linear operator between two Hausdorff TVSs. If   is a dense vector subspace of   and if the restriction   to   is a topological homomorphism then   is also a topological homomorphism.[3]

So if   and   are Hausdorff completions of   and   respectively, and if   is a topological homomorphism, then  's unique continuous linear extension   is a topological homomorphism. (However, it is possible for   to be surjective but for   to not be injective.)

Open mapping theorem

edit

The open mapping theorem, also known as Banach's homomorphism theorem, gives a sufficient condition for a continuous linear operator between complete metrizable TVSs to be a topological homomorphism.

Theorem[4] — Let   be a continuous linear map between two complete metrizable TVSs. If   which is the range of   is a dense subset of   then either   is meager (that is, of the first category) in   or else   is a surjective topological homomorphism. In particular,   is a topological homomorphism if and only if   is a closed subset of  

Corollary[4] — Let   and   be TVS topologies on a vector space   such that each topology makes   into a complete metrizable TVSs. If either   or   then  

Corollary[4] — If   is a complete metrizable TVS,   and   are two closed vector subspaces of   and if   is the algebraic direct sum of   and   (i.e. the direct sum in the category of vector spaces), then   is the direct sum of   and   in the category of topological vector spaces.

Examples

edit

Every continuous linear functional on a TVS is a topological homomorphism.[1]

Let   be a  -dimensional TVS over the field   and let   be non-zero. Let   be defined by   If   has it usual Euclidean topology and if   is Hausdorff then   is a TVS-isomorphism.

See also

edit

References

edit
  1. ^ a b c d e f g h Schaefer & Wolff 1999, pp. 74–78.
  2. ^ Köthe 1969, p. 91.
  3. ^ a b Schaefer & Wolff 1999, p. 116.
  4. ^ a b c Schaefer & Wolff 1999, p. 78.

Bibliography

edit
  • Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190.
  • Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
  • 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.
  • Köthe, Gottfried (1979). Topological Vector Spaces II. Grundlehren der mathematischen Wissenschaften. Vol. 237. New York: Springer Science & Business Media. ISBN 978-0-387-90400-9. OCLC 180577972.
  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
  • Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
  • Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
  • Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067.
  • 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.
  • Valdivia, Manuel (1982). Nachbin, Leopoldo (ed.). Topics in Locally Convex Spaces. Vol. 67. Amsterdam New York, N.Y.: Elsevier Science Pub. Co. ISBN 978-0-08-087178-3. OCLC 316568534.
  • Voigt, Jürgen (2020). A Course on Topological Vector Spaces. Compact Textbooks in Mathematics. Cham: Birkhäuser Basel. ISBN 978-3-030-32945-7. OCLC 1145563701.
  • Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.