## Distance between vector and closed subspace

Post any maths and/or physics related questions here.

Moderators: Darobat, RecursiveS, Dante Shamest, Bugdude, Wizard

### Distance between vector and closed subspace

Hi everyone. I'm new here and this is my first post. While I'm sure the problem isn't so complicated, I've spent with it two weeks with no success, so I would be glad to get any help.
I need help with following problem :
X is Hilbert vector space, Y is closed subspace of X, d:X -> R is a distance function, that means: for any x in X ,d(x)=inf{ ||x-y||, y in Y}
I need to prove that for any x in X there is y in Y such that d(x)=||x-y||.
The proof is quite simple if X=R^n; but my X isn't necessarily euclidian and its dimension isn't necessarily finite; so I can't use assumption that closed and bounded set is compact.
A lot of thanks for any response.
ttttolik

Posts: 2
Joined: Mon Jun 08, 2009 8:16 am

### Re: Distance between vector and closed subspace

Hilbert spaces were essentially not covered in my Math education, so it's unlikely I'll be able to help much. However, it seems that it's easy to find information about this problem online: http://en.wikipedia.org/wiki/Hilbert_space#Best_approximation (a linear subspace is obviously convex, so you are covered by the conditions)

Alvaro
Moderator

Posts: 5185
Joined: Mon Sep 22, 2003 4:57 pm
Location: NY, USA

### Re: Distance between vector and closed subspace

Thanks a lot for the link; the problem is solved.By the way, while chasing the beast I've met a book that should provide answers to much wider space of question on the subject. It's "Best approximation in inner product spaces" by Frank Deutsch - for everyone concerned.
ttttolik

Posts: 2
Joined: Mon Jun 08, 2009 8:16 am

### Re: Distance between vector and closed subspace

Join to the thanks
I'm new here, but I've found so many useful things here
very greatful to you for your work
Though some posts are quite old, they are still helpful
jennyfer99

Posts: 4
Joined: Fri Sep 02, 2011 5:19 am