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

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@hausdorffm Жыл бұрын

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.

@akrishna1729 2 жыл бұрын

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.

@shivangsachar5144 Жыл бұрын

I don't understand why convexity is necessary. If you take the complement of the open ball, then the infimum is attained on the boundary, which is in the set (sure, it's not unique). Can you give an example of a convex set where the infimum is not attained anywhere?

@akrishna1729 Жыл бұрын

@Shivang Sachar I believe this is to demonstrate failure of uniqueness specifically. I'm not sure if convexity is needed for existence, but I will try to come up with/search for a proof or a counterexample.

@arkarupbasumallik Ай бұрын

@@akrishna1729don't you need convexity to show that (u_{n}+u_{m}) /2 is in C for the proof of the existence part while establishing that the candidate sequence is Cauchy?