1
$\begingroup$

Let H be the Hilbert space L$^2$([0,1)], and let $S$ be the subspace of functions f $\in$ H satisfying $\int^1_0(1+x)f(x)dx=0$.

Find the element of $S$ closest to the function $g\in H$ defined by $g(x)=1$. Prove that it is the closest element.

I think what I need to do is find the orthogonal projection of $g$ onto $S$. I'm not exactly sure how I would go about doing that. Any help would be appreciated.

$\endgroup$
1
$\begingroup$

Note that $$S=(1+x)^\perp$$ Thus $S$ is closed and so we know, by the projection theorem, that there's a unique $f\in S$ such that $||f-g||=\min_{h\in S}\{||h-g||\}$. Furthermore, this $f$ is the unique element of $S$ satisfying $(f-g)\perp S$. But $$S^\perp = ((1+x)^\perp)^\perp=\overline{(1+x)}=1+x$$ so we can solve for $f$.

$\endgroup$
  • $\begingroup$ Great answer, although (minor objection) it seems confusing to me to not distinguish between $1+x$ and the subspace generated by $1+x$. You're also not distinguishing between the function $x \mapsto 1+x$ and the expression $1+x$, but I think that's more common and excusable. $\endgroup$ – 6005 Aug 27 '16 at 0:15

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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