In a reflexive Banach space using the weakly compact unite ball to show minimal distance

Asked Dec 06 '20 at 06:54

Active Dec 06 '20 at 07:20

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Let $X$ be a Banach space, and $Y$ be a closed subspace of $X$. Suppose the closed unit ball $B(0,1)=\{x\in X : \|x \| \leq 1 \}$ is weakly compact in $X$. Show: given $x \in X$, there is a point in $Y$ of minimal distance to $x$.

EDIT: Attempt:

Using weakly compactness of the unit ball, we know that $X$ is reflexive.

Now I know that since $Y$ is convex and closed, it also weakly closed; so that I construct a minimizing sequence $(y_n)$ in $Y$ such that $\|x- y_n \|$ will converge to the minimal distance.

But I'm not sure how to show that $\|x- y_n \|$ will converge to the minimal distance.

I feel like I could somehow use lower semicontinuity to get a minimal distance. But then in this case, $Y$ has to be compact.

Also, I haven't used that $X$ is reflexive...

Any help is appreciated!

Thanks!

edited Dec 06 '20 at 07:20

asked Dec 06 '20 at 06:54

Toasted_Brain

$Y$ is a subspace, right? It's definitely convex! Every convex combination is a linear combination. – user854214 Dec 06 '20 at 06:59
RIGHT! How could I miss that? Thanks! – Toasted_Brain Dec 06 '20 at 07:08
Does this help you get further with your attempt? $Y$ isn't (necessarily) compact, but it's boundedly compact. – user854214 Dec 06 '20 at 07:46
Also, there's an answer here, which looks to be very much along the lines of your own attempt. – user854214 Dec 06 '20 at 07:57
Wow! Yes! I think this might be it. I am going to have a closer look. But this link totally saves my day! Thanks! – Toasted_Brain Dec 06 '20 at 08:04

In a reflexive Banach space using the weakly compact unite ball to show minimal distance

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