McShane's Lemma

The lemma McShane , English McShane's lemma , is a theorem that between the mathematical areas of general topology and functional analysis is based. The lemma goes back to the US mathematician Edward James McShane and deals with the question of the continuation of Lipschitz continuous real-valued functions on subspaces of metric spaces .

Formulation of the lemma

The lemma says:

Let be a metric space, be a subspace located in it and be ${\displaystyle X}$ ${\displaystyle A\subseteq X}$

${\displaystyle f\colon A\rightarrow \mathbb {R} }$

a Lipschitz continuous real-valued function with the Lipschitz constant . ${\displaystyle A}$ ${\displaystyle L>0}$

Then:

${\displaystyle f}$ has a continuous continuation

${\displaystyle F\colon X\rightarrow \mathbb {R} }$

with the same Lipschitz constant . ${\displaystyle L}$

Related sentence

A related theorem is Kirszbraun's theorem , which deals with the same question in the context of the Euclidean (or Hilbert space ) and comes to the same result, albeit under different conditions. Neither of the two results directly implies the other. However, they overlap in the event that there is a (or a Hilbert space) and a subset and a Lipschitz continuous mapping are also taken as a basis here. ${\displaystyle X}$ ${\displaystyle \mathbb {R} ^{n}\;(n\in \mathbb {N} )}$ ${\displaystyle A\subseteq X}$ ${\displaystyle f\colon A\rightarrow \mathbb {R} ^{m}}$ ${\displaystyle m=1}$

literature

• Philippe G. Ciarlet : Linear and Nonlinear Functional Analysis with Applications . Society for Industrial and Applied Mathematics, Philadelphia PA 2013, ISBN 978-1-61197-258-0 ( MR3136903 ).
• Edward J. McShane : Extension of range of functions . In: Bulletin of the American Mathematical Society . tape 40 , 1934, pp. 837-842 , doi : 10.1090 / S0002-9904-1934-05978-0 ( MR1562984 ).

References and footnotes

1. ^ ^{A b} E. J. McShane: Extension of range of functions. 'n: Bulletin of the American Mathematical Society. Volume 40, 1934, pp. 837-842.
2. ^ ^{A b} Philippe G. Ciarlet: Linear and Nonlinear Functional Analysis with Applications. 2013, pp. 154–155