# If $V\subset L^{p}(\mathbb{R})$ is closed, then the distance from $f$ to $V$ is the distance from $f$ to some point in $V$

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.

• 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. – Android Netizen Sep 5 '16 at 22:36
• 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? – user251257 Sep 5 '16 at 22:45
• Banach Alaouglu says it is weak* compact – Android Netizen Sep 5 '16 at 22:46
• @AndroidNetizen there are ... for reason – user251257 Sep 5 '16 at 22:47
• @AndroidNetizen if the Banach space is reflexiv, then as corollary of Alaoglu every normed bounded sequence has a weakly convergent subsequence – 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 $$\tag{*}d(f,V)\leqslant \left\lVert f-v_n\right\rVert_p\lt d(f,V)+\frac 1n.$$
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 $$\lVert w\rVert_p\leqslant \liminf_{k\to +\infty}\lVert w_k\rVert_p$$.
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 $$v\in V$$.