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In a paper that I am reading there is a following step:

Let $X$ be a Banach space and let $(x_k) \subset X$ be a normalized sequence that converges weakly to $0$. Then $\overline{co}(x_k)$ is a weakly compact set.

(notice that $\overline{co}(x_k)$ denotes the norm-closure of the convex hull of $(x_k)$.)

I think that I managed to prove the claim, but I had to do a lot of manual checking. My line of thought is given below.

My question is: Can this be proved more directly than I did (assuming that my proof is without error)? For example, is closed convex hull of a weakly compact set always weakly compact? (similar question was asked here, but in a bit different context, so it does not seem to apply to this situation: Convex hulls of compact sets).

I reasoned as follows:

  • $\{x_k \mid k \in \mathbb{N} \}$ is an weakly compact set.
  • I checked that given a family $(y_{\alpha}) \subset co(x_k)$, it contains an weakly convergent subfamily, which weakly converges to an element in $\{ \sum_{k=1}^{\infty} {\alpha_k x_k \mid (\alpha_k) \in B_{\ell_1} }\}$.
  • I observed that given any family $(y_{\alpha})_{\alpha \in I} \subset \overline{co}(x_k)$, I can construct a family $(z_{\alpha, \epsilon})_{\alpha \in I, \epsilon > 0} \subset {co}(x_k)$ such that $\forall \alpha \in I, \epsilon > 0$ we have that $\lVert y_{\alpha} - z_{\alpha, \epsilon} \rVert < \epsilon$. By defining order for family $z_{\alpha, \epsilon}$ in such a way that $(\alpha_1, \epsilon_1) \leq (\alpha_2, \epsilon_2) \Leftrightarrow \alpha_1 \leq \alpha_2$ and $\epsilon_1 \geq \epsilon_2$, I can verify that (assuming that I did not make a mistake): \begin{equation*} (y_{\alpha}) \text { converges weakly to } w \Leftrightarrow (z_{\alpha, \epsilon}) \text{ converges weakly to } w \end{equation*}

  • Since $(y_{\alpha}) \subset \overline{co}(x_k)$, then $(z_{\alpha, \epsilon})_{\alpha \in I, \epsilon > 0} \subset {co}(x_k)$. The latter family contains an weakly convergent subfamily $(z_{\beta}')$, which converges to an element $c \in \{ \sum_{k=1}^{\infty} {\alpha_k x_k \mid (\alpha_k) \in B_{\ell_1} }\} \subset \overline{co}(x_k)$. Therefore the original family $(y_{\alpha}) \subset \overline{co}(x_k)$ can also be shown to have a subfamily $(y_{\gamma}')$ which converges weakly to the same element $c$.

  • Therefore $\overline{co}(x_k)$ is a weakly compact set.

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  • $\begingroup$ What paper is it? $\endgroup$ – Francois Ziegler Sep 12 '15 at 22:34
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    $\begingroup$ In the statement of the result you need to change to `weakly compact' $\endgroup$ – Mikhail Ostrovskii Sep 12 '15 at 23:32
  • $\begingroup$ @FrancoisZiegler, indeed, I meant weakly compact. Sorry for the mistake. $\endgroup$ – Rauni Sep 14 '15 at 12:26
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    $\begingroup$ @MikhailOstrovskii, here is the paper: math.slu.edu/~freeman/wGschurJFA.pdf . It is called "A weak Grothendieck compactness principle". $\endgroup$ – Rauni Sep 14 '15 at 12:27
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    $\begingroup$ @Rauni You may then be interested in this master's thesis I came across. It gives an exposition of the Freeman et al. paper, and in particular a proof of your desired claim: see Remark 1 following Lemma 1.3. $\endgroup$ – Francois Ziegler Sep 14 '15 at 15:24
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You can find in many functional analysis text books the theorem that the closed convex hull of a weakly compact subset of a Banach space is weakly compact. But what you want is simpler than the general theorem. Here is a simple conceptual proof: Let $(y_n)$ be a weakly null sequence in $X$ and consider the bounded linear operator $T:\ell_1 \to X$ that maps the $n$th unit vector in $\ell_1$ to $y_n$. By Banach-Alaoglu, to show that the closed convex hull of $(y_n)$ is weakly compact it is sufficient to verify that $T$ is weak$^*$ to weak continuous. But that is the same as saying that $T^*$ maps $X^*$ into the predual $c_0$ of $\ell_1$, which in turn is the same as saying that $(y_n)$ converges weakly to zero.

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  • $\begingroup$ Thank You for Your explanation. I need to think about this for a bit. I will mark Your answer as accepted as soon as I am able. $\endgroup$ – Rauni Sep 14 '15 at 13:44
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Rauni’s link to the Freeman et al paper perhaps provides sufficient excuse for me to give here a simple proof (which, however, uses some theory) of the theorem in that paper. The background can more or less be found in Diestel’s book “Sequences and Series in Banach Spaces”, Springer GTM 92. The theorem says that if every weakly compact subset of the Banach space $X$ is in the closed convex hull of a weakly null sequence, then $X$ is a Schur space (which means that every weakly convergent sequence in $X$ is norm convergent). If $X$ has the property but is not Schur, then there is a normalized weakly null sequence $(y_n)$ in $X$ that, by passing to a subsequence, can assumed to be basic. As in my other answer to this MO problem, let $T:\ell_1 \to X$ be the weak$^*$ to weak continuous bounded linear operator that maps the $n$th unit vector in $\ell_1$ to $y_n$. The operator $T$ is weakly compact but not compact, and $T$ is injective because $(y_n)$ is basic. By the factorization theorem for weakly compact operators, $T= JS$ where $S:\ell_1 \to R$, $J:R\to X$ are injective bounded linear operators and $R$ is reflexive. We can now forget about $T$ and focus on $J$. Necessarily $J$ is not compact. But the image $JB_R$ of the closed unit ball $B_R$ of $R$ is weakly compact, so by hypothesis is in the closed convex hull of a weakly null sequence $(x_n)$ in $X$. Again, the sequence $(x_n)$ is the image under a weak$^*$ to weak continuous bounded linear operator $U: \ell_1 \to X$ that maps the $n$th unit vector in $\ell_1$ to $x_n$. If $U$ were injective we would be done since $U^{-1}J$ would be a necessarily non compact bounded linear operator from a reflexive space into $\ell_1$, which is impossible because $\ell_1$ has the Schur property. The same contradiction is available as long as $\ell_1/U^{-1}(0)$ has the Schur property. But since $U$ is weak$^*$ to weak continuous, the kernel $U^{-1}(0)$ of $U$ is weak$^*$ closed in $\ell_1$, which is to say that $\ell_1/U^{-1}(0)$ is the dual of the subspace $U^{-1}(0)_\perp$ of $c_0$. But the dual of every subspace of $c_0$ has the Schur property (proofs that $\ell_1$ has the Schur property also give this).

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  • $\begingroup$ When You say "By the factorization theorem for weakly compact operators", do You refer to the DFJP Factorization or to some other result? $\endgroup$ – Rauni Oct 1 '15 at 21:46
  • $\begingroup$ Yes, DFJP, Rauni. $\endgroup$ – Bill Johnson Oct 2 '15 at 20:04
  • $\begingroup$ Thank You for Your alternate proof. I used it in a seminar. But it seems to me that we do not use injectivity of J anywhere? $\endgroup$ – Rauni Oct 6 '15 at 5:08
  • $\begingroup$ True, Rauni, but the proof of the factorization theorem gives an injective $J$. Also, if you have such a factorization with a non injective $J$, you can mod out the kernel of $J$ to get a factorization through another reflexive space for which the mapping from the reflexive space is injective. $\endgroup$ – Bill Johnson Oct 7 '15 at 7:08

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