An example of non complete inner product space is given by with the inner product .
An example of complete non empty convex subset is given by a finite dimensional subspace (it's works in each inner product space).
The best approximation theorem states that if (H,<,>) is an inner product space and M is a non-empty, complete, convex subset of H, then for every x in H there is a unique y in M such that d(x,M) = ||y-x||.
I don't know any good examples of inner product spaces (especially not complete IPS) with a complete convex subset. Any ideas?
Thanks!
An example of non complete inner product space is given by with the inner product .
An example of complete non empty convex subset is given by a finite dimensional subspace (it's works in each inner product space).