2
$\begingroup$

I am looking at the Hilbert Projection Theorem, which states that every non-empty closed convex set in a Hilbert space admits a unique element that has the minimum norm in the set.

The proof involves taking the infimum of the norm on the set and constructing a seqeuence in the set with decreasing norm. However, this proof does not give us an explicit criterian or method to find a minimum element norm.

Finding this norm is easy in the cases when the closed convex set is a subspace or a sphere, but it may not be clear in other cases. For example if $H,G$ are Hilbert spaces and $A:H \rightarrow G$ is a bounded linear map, then for a closed convex set $C$ in $G$, $A^{-1}(C)$ may be a very strange set (depending on how $A$ is defined). In this case, are there any better results than the Hilbert Projection Theorem?

$\endgroup$
2
  • $\begingroup$ How are you defining $C$? $\endgroup$ Mar 13, 2018 at 23:59
  • $\begingroup$ I was meaning to give an example of a setting. I did not have any particular C in mind. My intention was to ask for general results in this direction. $\endgroup$ Mar 14, 2018 at 11:46

1 Answer 1

2
$\begingroup$

A constructive equivalent might be: Any located closed convex set has a unique element of minimal norm. This is roughly the version in chapter 7, problem 11 of Bishop and Bridges’s 1985 Constructive Analysis.

Here, locatedness is a standard constructive assumption, meaning that the distance to the set exists, or is calculable. Unpacking further, this means that for any $p$, there is a sequence $s_n$ of points in $S$ where the distances $d(p,s_n)$ converge to a number less than or equal to any $d(p,s)$. Then, as the Wikipedia article on the theorem shows, the $s_n$ are a Cauchy sequence and their limit is the desired point.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.