Skip to main content Accessibility help
×
Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-28T16:04:47.707Z Has data issue: false hasContentIssue false

1 - Calculus in Locally Convex Spaces

Published online by Cambridge University Press:  08 December 2022

Alexander Schmeding
Affiliation:
Nord Universitet, Norway
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.

Summary

It is well known that multidimensional calculus, aka Fréchet calculus, carries over to the realm of Banach spaces and Banach manifolds. Banach spaces are often not sufficient for our purposes. To generalise derivatives, we will, as a minimum, need vector spaces with an amenable topology (which need not be induced by a norm). This chapter presents first a notion of calculus in locally convex spaces, which requires the existence and continuity of directional derivatives. The resulting calculus is called Bastiani calculus and we compare it to some common (but inequivalent) notions of calculus such as the convenient calculus. Building on the chain rule, we then construct the basic building blocks of (infinite-dimensional) differential geometry: manifolds and their tangent spaces as well as submersions and immersions.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2022
Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NCCreative Common License - ND
This content is Open Access and distributed under the terms of the Creative Commons Attribution licence CC-BY-NC-ND 4.0 https://creativecommons.org/cclicenses/

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] 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.

Available formats
×

Save book to Dropbox

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

Available formats
×

Save book to Google Drive

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

Available formats
×