Continuation of Dugundji

The continuation of Dugundji ( Dugundji extension theorem or Dugundji extension formula ) is a mathematical theorem that is located in the transition field between general topology and the theory of topological vector spaces . It goes back to a scientific publication by the American mathematician James Dugundji from 1951 and is directly linked to Tietze-Urysohn's theorem about the continuation of continuous mappings of normal spaces , of which it represents a generalization in a certain sense .

Formulation of the sentence

The sentence can be formulated as follows:

A metric space and a closed subset as well as a locally convex topological vector space are given . ${\displaystyle X}$ ${\displaystyle A\subseteq X}$ ${\displaystyle L}$

Then there is a continuous continuation for each continuous mapping , that is, a continuous mapping with , which is such that the image area is encompassed by the convex envelope of . ${\displaystyle f\colon A\rightarrow L}$ ${\displaystyle X}$ ${\displaystyle F\colon X\rightarrow L}$ ${\displaystyle F|_{A}=f}$ ${\displaystyle F(X)}$ ${\displaystyle f(A)}$

In a slightly modified, but equivalent form, the continuation of Dugundji can also be represented as follows:

A metric space and a closed subset therein as well as a locally convex topological vector space and a convex subset therein are given . Furthermore, let it be a continuous mapping. ${\displaystyle X}$ ${\displaystyle A\subseteq X}$ ${\displaystyle L}$ ${\displaystyle K\subseteq L}$ ${\displaystyle f\colon A\rightarrow K}$

Then has a steady continuation . ${\displaystyle f}$ ${\displaystyle F\colon X\rightarrow K}$

Classification of the sentence

The Tietze-Urysohn extension theorem guarantees for normal topological spaces alone the existence of a steady continuation in the event that the range of values of the underlying continuous mapping one of intervals of composite product space , about one is. The continuation of Dugundji's theorem now provides a considerable expansion of this statement, which, however, is only possible when a metric space is used instead of a normal topological space : the generalization of the range of values ​​in Dugundji's theorem is bought at the price of specializing the domain of definition. ${\displaystyle K}$ ${\displaystyle f\colon A\rightarrow K}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle {\mathbb {R} }^{n}}$ ${\displaystyle X}$ ${\displaystyle X}$

literature

Original work

• James Dugundji : An extension of Tietze's theorem . In: Pacific Journal of Mathematics . tape 1 , no. 3 , 1951, ISSN  0030-8730 , p. 353-367 ( projecteuclid.org MR0044116 [PDF]).

Monographs

• Czesław Bessaga , Aleksander Pełczyński : Selected Topics in Infinite-dimensional Topology (=  Monograph Matematyczne . Volume 58 ). Państwowe Wydawnictwo Naukowe, 1975, ISSN  0077-0507 ( MR0478168 ).
• Karol Borsuk : Theory of Retracts (=  Monograph Matematyczne . Volume 44 ). Państwowe Wydawnictwo Naukowe, Warsaw 1967 ( MR0216473 ).
• James Dugundji : Topology . 8th edition. Allyn and Bacon , Inc., Boston, MA 1973.
• Andrzej Granas, James Dugundji: Fixed Point Theory (=  Springer Monographs in Mathematics ). Springer, New York NY a. a. 2003, ISBN 0-387-00173-5 ( MR1987179 ).
• Karl Heinz Mayer: Algebraic Topology . Birkhäuser, Basel a. a. 1989, ISBN 3-7643-2229-2 ( MR0974296 ).
• Horst Schubert : Topology (=  mathematical guidelines ). 4th edition. BG Teubner, Stuttgart 1975, ISBN 3-519-12200-6 ( MR0423277 ).

Web link

• Original work by Dugundji

Individual evidence

1. ↑ Dugundji: An extension of Tietze's theorem. 1951, pp. 353-367, here p. 353 ff.
2. ^ Bessaga, Pełczyński: Selected Topics in Infinite-dimensional Topology. 1975, p. 57 ff.
3. ↑ Granas, Dugundji: Fixed Point Theory. 2003, pp. 163-164.
4. ^ Mayer: Algebraic Topology. 1989, p. 56.
5. ↑ Dugundji: An extension of Tietze's theorem. 1951, pp. 353-367, here p. 357.
6. ↑ Borsuk: Theory of Retracts. 1967, pp. 77-78.
7. ^ Mayer: Algebraic Topology. 1989, pp. 54, 56
8. ↑ Dugundji: Topology. 1973, p. 189.
9. ^ Schubert: Topology. 1975, p. 83.
10. ^ Mayer: Algebraic Topology. 1989, p. 56.