2 edition of **On the extension of Lipschitz maps.** found in the catalog.

On the extension of Lipschitz maps.

Sten Olof SchoМ€nbeck

- 317 Want to read
- 29 Currently reading

Published
**1967**
by Almqvist & Wiksell in Stockholm
.

Written in English

- Banach spaces.,
- Topology.,
- Mappings (Mathematics)

**Edition Notes**

Bibliography: p. 209.

Series | Arkiv för matematik,, bd. 7, nr. 15 |

Classifications | |
---|---|

LC Classifications | QA3 .A7 bd. 7, nr. 15 |

The Physical Object | |

Pagination | 201-209 p. |

Number of Pages | 209 |

ID Numbers | |

Open Library | OL4505601M |

LC Control Number | 79350680 |

If U is a subset of the metric space M and f: U → R is a Lipschitz continuous function, there always exist Lipschitz continuous maps M → R which extend f and have the same Lipschitz constant as f (see also Kirszbraun theorem). An extension is provided by \tilde f(x):=\inf_{u\in U}\{ f(u)+k\, d(x,u)\}, where k is a Lipschitz constant for f on U. If U is a subset of the metric space M and f: U → R is a Lipschitz continuous function, there always exist Lipschitz continuous maps M → R which extend f and have the same Lipschitz constant as f (see also Kirszbraun theorem). An extension is provided by \tilde f(x):=\inf_{u\in U}\{ f(u)+k\, d(x,u)\}, where k is a Lipschitz constant for f.

You can write a book review and share your experiences. Other readers will always be interested in your opinion of the books you've read. Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. SPECTRAL CALCULUS AND LIPSCHITZ EXTENSION 5 right hand side of (4), but as will become clear from the ensuing con-siderations, De nition su ces for many purposes. In what follows, whenever we say that a metric space is W pbarycen-tric with constant and also q-barycentric with constant Kwe mean.

Abstract We consider Lipschitz maps with values in quasi-metric spaces and extend such maps to finitely many points. We prove that in this context every 1-Lipschitz map admits an extension such that its Lipschitz constant is bounded from above by the number of added points plus one. Moreover, we prove that if the source space is a Hilbert space and the target space is a Banach Cited by: 1. EXTENSION OF LIPSCHITZ FUNCTIONS another extension which is conceptually related to McShane’s one [6]. The addition of the cone X = { (x,~) E E X R,,U > k Ilxll} to the epigraph of $ gives, geometrically, an idea of what should be the epigraph of a possible extension. Concretely, some applications of Lipschitz extensions of functions to machine learning can be found in [5, 8, 9]. General explanations about applications of mathematical analysis in Machine Learning can be found in ; in particular, basic definitions, examples and results on Lipschitz maps can be found in Section of this book.

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SyntaxTextGen not activatedPdf Implicit Function Theorem for Lipschitz Maps A map f: X!Y is Lipschitz pdf there is a constant C such that for all x 1;x 2 2X, d Y (f(x 1);f(x 2)) Cd X(x 1;x 2). Every di erentiable map from an open set in R n to Rp is locally Lipschitz, but the converse is not true.

For example, the function f(x) = jxjis Lipschitz but not di erentiable at 0.The object of this book is a presentation of the major results relating to two geometrically inspired problems in analysis.

One download pdf that of determining which metric spaces can be isometrically embedded in a Hilbert space or, more generally, P in an L space; the other asks for conditions on a pair of metric spaces which will ensure that every contraction or every Lipschitz-Holder map from a.Bull. Amer.

Math. Soc. Vol Number 2 (), On the extension of Lipschitz, Lipschitz-Hölder continuous, and monotone functionsCited by: