1
$\begingroup$

This a qualifying exam question. Let $1<p<\infty$ and $V\subset L^p(\mathbb{R})$ be closed. Define

$$d(f,V) = \inf_{v\in V} \|f-v\|_p.$$

Prove there exists $v_0\in V$ such that $d(f,v)=\|f-v_0\|_p$. The hint is to consider a minimizing sequence $v_n$ and extract a weakly convergent subsequence.

Well, since $d(f,V)$ is the infimum of a set of numbers, can extract a sequence $v_n\in V$ such that $\|f-v_n\|_p$ converges to $d(f,V)$. However I am stuck at this point. Any help would be greatly appreciated.

$\endgroup$
  • $\begingroup$ Might not be - I was thinking of a fixed f - so I can separate the point f by a linear functional if needed. Sorry for the confusion. $\endgroup$ – Android Netizen Sep 5 '16 at 22:36
  • 2
    $\begingroup$ The hint says something about weakly convergent. I would concentrate on that. What do know about the ball in weak topology in case of a ... Banach space? $\endgroup$ – user251257 Sep 5 '16 at 22:45
  • $\begingroup$ Banach Alaouglu says it is weak* compact $\endgroup$ – Android Netizen Sep 5 '16 at 22:46
  • $\begingroup$ @AndroidNetizen there are ... for reason $\endgroup$ – user251257 Sep 5 '16 at 22:47
  • 1
    $\begingroup$ @AndroidNetizen if the Banach space is reflexiv, then as corollary of Alaoglu every normed bounded sequence has a weakly convergent subsequence $\endgroup$ – user251257 Sep 5 '16 at 23:03
1
$\begingroup$

By definition of infimum, for all integer $n$, there exists a function $v_n\in V$ such that $$\tag{*}d(f,V)\leqslant \left\lVert f-v_n\right\rVert_p\lt d(f,V)+\frac 1n.$$

We use the following facts about weakly convergent sequences:

  1. In a reflexive Banach space, every bounded sequence admits a weakly convergent subsequence.
  2. For $1\lt p\lt +\infty$, $L^p(\mathbb R)$ is reflexive.
  3. If $(w_k)_{k\geqslant 1}$ weakly converges in $L^p(\mathbb R)$ to some $w$, then $\lVert w\rVert_p\leqslant \liminf_{k\to +\infty}\lVert w_k\rVert_p$.
  4. A convex set closed for the topology of the norm is also closed for the weak topology.

Observe that $\lVert v_n\rVert_p\leqslant \lVert f-v_n\rVert_p+\lVert f\rVert_p\leqslant d(f,V)+1+\lVert f\rVert_p$. By facts 1. and 2., we can extract a convergent subsequence, denoted $(v_{n_k})_{k\geqslant 1}$, which converges to some $v$. Let $w_k:=v_{n_k}-f$. Then $w_k\to v-f=:w$ weakly hence by 3. and (*), $\left\lVert v-f\right\rVert_p=d(f,V)$. By 4. and the fact that $v_{n_k}\in V$, we have $v\in V$.

$\endgroup$

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.