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.

  • $\begingroup$ 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
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    $\begingroup$ @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

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 $v\in V$.


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