Continuation lemma

The Fortsetzungslemma ( English Extension lemma ) is a theorem that the transition field of the two mathematical subregions topology and functional analysis is attributable. The lemma deals with the fundamental question of the continuation of continuous functionals on certain topological spaces and is therefore related to (and even a consequence of) Tietze's continuation theorem .

formulation

The lemma can be formulated as follows:

Let a completely regular topological space and a compact subset be given . ${\displaystyle X\neq \emptyset }$ ${\displaystyle T\subseteq X}$

Let there be and the associated function spaces of the continuous functionals from or into the basic field , which should be either the field of real numbers or the field of complex numbers , each provided with the topological structure generated by the absolute value function . ${\displaystyle C(X,\mathbb {K} )}$ ${\displaystyle C(T,\mathbb {K} )}$ ${\displaystyle X}$ ${\displaystyle T}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle |\cdot |}$

Then:

For every functional there is a functional with ${\displaystyle f\in C(T,\mathbb {K} )}$ ${\displaystyle F\in C(X,\mathbb {K} )}$

(i) . ${\displaystyle F|_{T}=f}$

(ii) . ${\displaystyle \sup _{t\in T}{|f(t)|}=\sup _{x\in X}{|F(x)|}}$

Evidence sketch

In the situation in question, one regards as a subspace of his Stone-Čech compactification and applies Tietze's continuation theorem, taking into account that a Hausdorff space is and as a compact subspace counts both from and from there to the closed sets . ${\displaystyle X}$ ${\displaystyle \beta {X}}$ ${\displaystyle X}$ ${\displaystyle T}$ ${\displaystyle X}$ ${\displaystyle \beta {X}}$

annotation

Klaus Jänich also uses the keyword lemma in his topology in connection with Tietze's continuation theorem by speaking of Tietze's extension lemma . However, the continuation lemma formulated above and the Tietzian extension lemma do not coincide.

literature

• Klaus Jänich : Topology (=  mathematical guidelines ). 8th edition. Springer-Verlag , Berlin, Heidelberg 2005, ISBN 978-3-540-21393-2 ( MR2262391 ).
• Hans Jarchow : Locally Convex Spaces (=  mathematical guidelines ). BG Teubner , Stuttgart 1981, ISBN 3-519-02224-9 ( MR0511737 ).
• Horst Schubert : Topology (=  mathematical guidelines ). 4th edition. BG Teubner, Stuttgart 1975, ISBN 3-519-12200-6 ( MR0423277 ).

Individual evidence

1. ↑ ^{a b c} Hans Jarchow: Locally Convex Spaces. 1981, p. 29
2. ↑ Horst Schubert: Topology. 1975, p. 61
3. ^ Klaus Jänich: Topology. 2005, p. 140 ff