MIT 18.100B Real Analysis, Spring 2025 - Lecture 21: Integrals and Derivatives under Uniform Convergence

MIT 18.100B Real Analysis, Spring 2025 Instructor: Tobias Holck Colding View the complete course: https://ocw.mit.edu/courses/18-100b-real-analysis-spring-2025/ YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP62Ie7F_tTAhhXoX5_Cl8meG We use Weierstrass M-test to prove continuity of power series inside the radius of convergence. We also show that C([a; b]) with the supremum norm is Cauchy complete. This will later play an important role when we establish existence and uniqueness of solutions to ordinary differential equations. Toward the end of the lecture we show that integrals and differentiation behave well under uniform convergence. License: Creative Commons BY-NC-SA More information at https://ocw.mit.edu/terms More courses at https://ocw.mit.edu Support OCW at http://ow.ly/a1If50zVRlQ We encourage constructive comments and discussion on OCW’s YouTube and other social media channels. Personal attacks, hate speech, trolling, and inappropriate comments are not allowed and may be removed. More details at https://ocw.mit.edu/comments.