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The Banach fixed point theorem, which is presented and proved in this video, gives a sufficient criterion for a function to have a fixed point. It is an important tool for proofs in various areas of mathematics, e.g. for the Picard-Lindelöf theorem or the implicit function theorem. The BRAND NEW book: http://weitz.de/GDM The NEW book: http://weitz.de/PP/ Iterated root extraction: • Why the nth root converges to one... Metric spaces: • Topological spaces as an abstraction of... Completeness of the real numbers: • Dedekind cuts: How to extract the re... Babylonian root extraction: • Babylonian root extraction / Heron... Brouwer's fixed point theorem: • Brouwer's fixed point theorem (Brouwer... In the playlist context: http://weitz.de/y/utvSi9SYn8E?list=PL... Chronological list: http://weitz.de/haw-videos/ The slightly different math textbook: http://weitz.de/KMFI/ Illustrations by Heike Stephan: / haiartandillustration General notes: http://weitz.de/youtube.html