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Let $H$ be a Hilbert space, $a \in H$ and $(x_n)_n$ a bounded sequence in $H$ such that every subsequence of $H$ that converges weakly converges to $a$. How do you prove that $(x_n)_n$ converges weakly to $a$? The only thing I know is that there is indeed at least one subsequence of $(x_n)_n$ that converges weakly.

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  • $\begingroup$ Something's wrong, or this is trivial. $\endgroup$ – Aweygan May 17 '18 at 4:25
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    $\begingroup$ It does not say that every subsequence is $w$-convergent, only that those that are, they converge to the same limit. $\endgroup$ – Theo May 17 '18 at 4:29
  • $\begingroup$ @Theo Oh I read this incorrectly. $\endgroup$ – Aweygan May 17 '18 at 4:31
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If $(x_n)_n$ does not converge weakly to $a$, then there is a subsequence of $(x_n)_n$ with the property that for some $v \in H$, there is some $\varepsilon > 0$ such that $|\langle x_n ,v \rangle - \langle a , v \rangle| \geq \varepsilon$ for every $n$ indexing the subsequence. But that subsequence is bounded and therefore admits a weakly convergent subsubsequence that weakly converges to $a$, which is a contradiction to the fact that the subsequence stayed away from $a$.

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  • $\begingroup$ Nice and simple. Thanks. $\endgroup$ – 97DL May 17 '18 at 4:41
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Note that every subsequence of $(x_n)_{n\in \mathbb{N}}$ is also a bounded sequence and therefore has a weakly convergent subsequence which converges to $a$ by hypothesis. As every subsequence of $(x_n)_{n\in \mathbb{N}}$ has a further subsequence that converges to $a$, $x_n\rightharpoonup a$.

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This works in general in reflexive Banach spaces. Let $y^*\in X^*$ be a bounded functional. We have that $y^{*}(x_n)$ is a bounded sequence, so if it is not convergent it must have at least two distinct accumulation points. Say $(y_n)$ and $(z_n)$ are subsequences of $(x_n)$ such that $y^{*}(y_n)\to \alpha$ $y^{*}(z_n)\to \beta$. Since $(y_n)$ and $(z_n)$ are bounded and $X$ is reflexive (so bounded sets are relatively weakly compact), each must contain a $w$-convergent subsequence. However any $w$-convergent subsequence converges to $a$, hence $\alpha=\beta=y^*(a)$. This shows that $y^{*}(x_n)$ is convergent, and it must converge to $y^*(a)$. Since $y^*$ was arbitrary, this shows $(x_n)$ is $w$-convergent.

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    $\begingroup$ Even better, this is true in any topology where boundedness implies sequential compactness. $\endgroup$ – Michael Lee May 17 '18 at 5:06
  • $\begingroup$ That is, relative sequential compactness (i.e. the sequential closure of a bounded set is sequentially compact). $\endgroup$ – Michael Lee May 17 '18 at 5:13
  • $\begingroup$ They are the same, by Eberlein-Smulian. $\endgroup$ – Theo May 17 '18 at 5:14
  • $\begingroup$ In Banach spaces, yes, but there are far more general settings where this is true. $\endgroup$ – Michael Lee May 17 '18 at 5:14
  • $\begingroup$ I see, your second comment clarified your first, not what I wrote in the proof. Of course, you are right. $\endgroup$ – Theo May 17 '18 at 5:17

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