Skip to main content Accessibility help
×
Home
Hostname: page-component-59b7f5684b-vh8gq Total loading time: 0.301 Render date: 2022-10-05T08:58:36.567Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": true, "useSa": true } hasContentIssue true

BORNOLOGIES AND LOCALLY LIPSCHITZ FUNCTIONS

Published online by Cambridge University Press:  15 May 2014

GERALD BEER*
Affiliation:
Department of Mathematics, California State University Los Angeles, 5151 State University Drive, Los Angeles, CA 90032, USA email gbeer@cslanet.calstatela.edu
M. I. GARRIDO
Affiliation:
Instituto de Matemática Interdisciplinar (IMI), Departamento de Geometría y Topología, Universidad Complutense de Madrid, 28040 Madrid, Spain email maigarri@mat.ucm.es
Rights & Permissions[Opens in a new window]

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\langle X,d \rangle $ be a metric space. We characterise the family of subsets of $X$ on which each locally Lipschitz function defined on $X$ is bounded, as well as the family of subsets on which each member of two different subfamilies consisting of uniformly locally Lipschitz functions is bounded. It suffices in each case to consider real-valued functions.

Type
Research Article
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Atsuji, M., ‘Uniform continuity of continuous functions of metric spaces’, Pacific J. Math. 8 (1958), 1116.CrossRefGoogle Scholar
Beer, G., Topologies on Closed and Closed Convex Sets (Kluwer Academic, Dordrecht, The Netherlands, 1993).CrossRefGoogle Scholar
Beer, G., ‘Embeddings of bornological universes’, Set-Valued Anal. 16 (2008), 477488.CrossRefGoogle Scholar
Beer, G. and Levi, S., ‘Total boundedness and bornologies’, Topology Appl. 156 (2009), 12711288.CrossRefGoogle Scholar
Bourbaki, N., Elements of Mathematics. General Topology. Part 1 (Hermann, Paris, 1966).Google Scholar
Caserta, A., Di Maio, G. and Holá, L., ‘Arzelà’s theorem and strong uniform convergence on bornologies’, J. Math. Anal. Appl. 371 (2010), 384392.CrossRefGoogle Scholar
Engelking, R., General Topology (Polish Scientific Publishers, Warsaw, 1977).Google Scholar
Garrido, M. I. and Jaramillo, J., ‘Homomorphisms on function lattices’, Monatsh. Math. 141 (2004), 127146.CrossRefGoogle Scholar
Garrido, M. I. and Jaramillo, J., ‘Lipschitz-type functions on metric spaces’, J. Math. Anal. Appl. 340 (2008), 282290.CrossRefGoogle Scholar
Garrido, M. I. and Meroño, A. S., ‘New types of completeness in metric spaces’, Ann. Acad. Sci. Fenn. Math. to appear.Google Scholar
Goldberg, R., Methods of Real Analysis, 2nd edn (Wiley, New York, 1976).Google Scholar
Hejcman, J., ‘Boundedness in uniform spaces and topological groups’, Czech. Math. J. 9 (1959), 544563.Google Scholar
Hogbe-Nlend, H., Bornologies and Functional Analysis (North-Holland, Amsterdam, 1977).Google Scholar
Roberts, A. and Varberg, D., Convex Functions (Academic Press, New York, 1973).Google Scholar
Taylor, A. and Lay, D., Introduction to Functional Analysis, 2nd edn (Wiley, New York, 1980).Google Scholar
Vroegrijk, T., ‘Uniformizable and realcompact bornological universes’, Appl. Gen. Topol. 10 (2009), 277287.CrossRefGoogle Scholar
You have Access
17
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

BORNOLOGIES AND LOCALLY LIPSCHITZ FUNCTIONS
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

BORNOLOGIES AND LOCALLY LIPSCHITZ FUNCTIONS
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

BORNOLOGIES AND LOCALLY LIPSCHITZ FUNCTIONS
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *