Convex Norms and Unique Best Approximations

In this video, we explore what it means for a norm to be convex. In particular we will look at how convex norms lead to unique best approximations.

For example, for any continuous function there will be a unique polynomial which gives the best approximation over a given interval.

Chapters
0:14 - Geometry of the Lp Norm
0:45 - Convexity of the Lp Norm
2:13 - Best Approximations are unique for convex norms (proof)
4:41 - Example

The product links below are Amazon affiliate links. If you buy certain products on Amazon soon after clicking them, I may receive a commission. The price is the same for you, but it does help to support the channel :-)

The Approximation Theory series is based on the book "Approximation Theory and Methods" by M.J.D. Powell:
https://amzn.to/3CXQpQZ

Errata and Clarifications:
The L1/2 "norm" isn't actually a norm! Lp for p less than 1 fail the triangle inequality.
All norms and normed linear spaces are convex, the important bit for the proof is that A must be a convex subset.
In the last section I show a graph labelled exp(-x) but that's not correct looking at the curve. It doesn't hurt the explanation, any curve would've been fine.

This video was made using:
Animation - Apple Keynote
Editing - DaVinci Resolve
Mic - Shure SM58 (with Behringer U-PHORIA UM2 audio interface)

Supporting the Channel.
If you would like to support me in making free mathematics tutorials then you can make a small donation over at
https://www.buymeacoffee.com/DrWillWood  
Thank you so much, I hope you find the content useful.

• #education
• #norm
• #lp_norm
• #vector_norm
• #normed_vector_space
• #normed_linear_space
• #approximation
• #best_approximation