The Hilbert Projection Theorem - Full Proof! (Theory behind Machine Learning) @JoelRosenfeld

Uploaded April 2024 | Updated September 2026, 1 day ago

In this video we embark on the study of spaces of (potentially) infinite dimension, Hilbert spaces. Here we give the full proof the Hilbert Projection theorem, which is the most critical piece of their study.

//Watch Next
The Real Analysis Survival Guide youtu.be/v5rD0B-zfXw
The Analyticity of the Laplace transform youtu.be/FIMkbFQL6XM
Introduction to Control Theory youtu.be/0v4WFmOm764

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The Hilbert Projection Theorem - Full Proof! (Theory behind Machine Learning)

Introduction to the Spectrum of Bounded Operators

$Laplace Transform of Derivatives of Distributions We talk about Distributional Derivatives and impact on the Laplace transform. Interestingly, this is a bit different than what happens for Laplace transforms of functions. //Additional Reading Zemanian - An Introduction to Generalized Functions and the Generalized Laplace and Legendre Transformations (JSTOR) https://www.jstor.org/stable/2027808?seq=1#metadata_info_tab_contents (SIAM) https://epubs.siam.org/doi/abs/10.1137/1010001?journalCode=siread //Exercises - Show that differentiation is a continuous operation on L. - Show that 1/n phi(t/n) e^{-st} goes to zero in L (i.e. with respect to each seminorm) as n goes to infinity for every s in C. //Watch Next The Schwartz Space (Full Lecture) https://youtu.be/StzEO8B1pFY Undergraduate Intro to the Laplace Transform https://youtu.be/mJSqt7nchRQ Introduction to Control Theory https://youtu.be/0v4WFmOm764 Fixed Point Iteration - Loki gets slapped 16 times for SCIENCE https://youtu.be/_I790SqNDjs //Music Provided by Epidemic Sound Vegas - Onda Norte Late Nights - Daxtan Twist - Cushy We Come Alive - Waycap Opposite - Jones Meadows Video Call from Las Angeles - Trevor Kowalski From the Cliffside - Tape Machines Cant Wait for You - Deanz Use this referral link to get a 30 day free trial with Epidemic Sound for your YouTube channel: https://www.epidemicsound.com/referral/644nao //Books Advanced Mathematical Analysis - Beals https://amzn.to/3fn2jdP From Vector Spaces to Function Spaces - Yutaka Yamamoto https://amzn.to/3sU2CVj Feedback Control Theory - Doyle, Francis, Tannenbaum https://amzn.to/3HFvuov Pick Interpolation and Hilbert Function Spaces - Agler and McCarthy https://amzn.to/3EOJU3K Nonlinear Systems - Khalil https://amzn.to/32Zyk8E Modern Control Systems - Dorf and Bishop https://amzn.to/331JQAm Fundamentals of Differential Equations - Nagle, Saff, and Snider https://amzn.to/3qeGD9C //Recording Equipment Canon SL3: https://amzn.to/3nZ11KU Canon T6i: https://amzn.to/3FUpkQh Rode VideoMic: https://amzn.to/3lhldGa Blue Yeti Microphone: https://amzn.to/3I1y88N Yeti Nano Microphone: https://amzn.to/3I1mriA SanDisc 256GB SD Card: https://amzn.to/3E3LOOr Neewer 5600K USB LED Lights: https://amzn.to/3xvB9cN Neewer 18 inch Ring Light: https://amzn.to/2ZvgCsc Camera Power Adapter: https://amzn.to/3D3upUu DISCLAIMER: The links above in this description may be affiliate links. If you make a purchase with the links provided I may receive a small commission, but with no additional charge to you :) Thank you for supporting my channel so that I can continue to produce mathematics content for you! 0:00 Introduction 0:48 About Laurent Schwartz 1:26 Derivatives of Regular Distributions 2:56 Autobiography of Laurent Schwartz 3:45 Derivatives of Distributions 5:11 Laplace Transforms of Derivatives 6:58 Too Simple yet Too Complicated 7:30 The Transfer Function$

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