# Math Help - Example for best approximation theorem

1. ## Example for best approximation theorem

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!

2. ## Re: Example for best approximation theorem

An example of non complete inner product space is given by $H:=\left\{f\in\mathcal{C}^1{\left[0,1\right]}, f(0=f(1)=0\right\}$ with the inner product $\langle f,g\rangle :=\int_0^1 f(t)g(t)dt+\int_0^1 f'(t)g'(t)dt$.
An example of complete non empty convex subset is given by a finite dimensional subspace (it's works in each inner product space).

3. ## Re: Example for best approximation theorem

So in this case, any polynomial space, Pn, for n finite.