In mathematics, the Dugundji extension theorem is a theorem in general topology due to American mathematician James Dugundji. It is directly related to the TietzeâÂÂUrysohn extension theorem â about extending continuous functions on normal spaces â of which it is, in a sense, a generalization.
Let be a metrizable space, a closed subset of X, and a locally convex topological vector space. Then:
or, equivalently:
The first version of the Tietze extension theorem corresponds to the special case of the above theorem where the target space L is the real line âÂÂ. Urysohn generalized this to replacing the domain being a metric space by an arbitrary normal space. The Dugundji extension theorem is a transverse generalization, replacing the target â by an arbitrary locally convex space. There is another generalization of the Tietze theorem assuming that the domain X is paracompact and the target L is a Banach space.
Fix some metric on Consider the open cover of that consists of the open balls for Since every metric space is paracompact, there exists a locally finite open cover of such that each is contained in one of those balls. Choose a partition of unity subordinate to this cover. For each , pick a point satisfying
which is possible since for each , there is an with . Define the extension on by:
The map is clearly continuous on . We shall then show it is continuous at each point in as well. For each in , we have: or
Thus, we have:
and then
Now, let a convex neighborhood C of be given. Then, since is continuous, there is some such that . Then we have by the above inequality, completing the proof of the continuity.
The article started as a machine (ChatGPT) translation of the corresponding article in French Wikipedia https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_prolongement_de_Dugundji.