Lecture 17: Minimizers, Orthogonal Complements and the Riesz Representation Theorem

MIT 18.102 Introduction to Functional Analysis, Spring 2021
Instructor: Dr. Casey Rodriguez

We move back to the general theory of Hilbert spaces with applications to concrete problems, discussing minimizers, orthogonal decomposition, projections, and one of the most fundamental “existence and uniqueness” results, the Riesz Representation Theorem!

License: Creative Commons BY-NC-SA

[hausdorffm]

At 6:57, both examples are same. I guess one of example should show that if we drop the convexity in the theorem, then the uniqueness no longer holds. In the first example, I guess what he want to show was that the minimizer is not unique, but the all points of the circle, so uniqueness does not holds when C is concave.

[akrishna]

First of all, thank you so much for this clear and lucid presentation of the material. At 7:30, when Dr. Rodriguez illustrates the example of the complement of an open ball in R^2, he states that the set is neither closed nor convex -- I believe he meant to say that it is only non-convex (it would be closed by definition, as the complement of an open set in the usual metric topology) which would demonstrate the necessity of convexity; the minimum norm occurs across the boundary of the disk. If it were both closed and convex as a counterexample, we couldn't conclude that either property was necessary for the length-minimization property to hold. To show that being closed is also needed, we could also (in addition to Dr. Rodriguez's second example) take an open ball not centered at the origin: a unique minimum-norm element (some point closest to zero) would occur on its boundary, which the ball does not contain because that is its set of limit points.

[ktayeb]

your inf is a minimum, because it is attained

[SSNewberry]

He is slow but packs a punch all the way to the end of the lecture.