Bill Johnson
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 2d comment Weakenings of the Bounded Approximation Property Yes, of course, Yemon. I'll blame it on wine. :) 2d comment Weakenings of the Bounded Approximation Property (1), (2), and (2)' are equivalent to the $\lambda$-BAP for the reasons you suggest. It is a good exercise to work this out if you are interested in the topic. (3) likely does not imply the others, but I don't have time to think about it now. Previous MO postings on the AP might have something relevant for that. Or maybe my paper with Oikhberg essentially contains an example--have you looked at that paper? Feb 4 answered p-summable sequence Feb 3 comment Characterization of Dedekind complete Riesz spaces by strictly positive functionals I think you can deduce what you want from Section 12 in "Positive Operators" by Aliprantis and Burkinshaw. Jan 13 answered Antiproximanal subspace of $L_1[0,1]$ Jan 13 comment Antiproximanal subspace of $L_1[0,1]$ The $Y$ in Mikhail's answer has codimension one. Obviously $Y$ cannot be reflexive, but $Y$ can be of any non zero finite codimension or of infinite codimension. (Let $Z$ be any separable Banach space and let $Q$ be an operator from $L_1$ to $Z$ that maps the closed unit ball of $L_1$ onto the open unit ball of $Z$. The kernel $Y$ of such a quotient map $Q$ is antiproximinal. It is easy to built such an operator from $\ell_1$ onto $Z$; to get one from $L_1$ compose the operator from $\ell_1$ with a norm one projection from $L_1$ onto a subspace that is isometric to $\ell_1$.) Jan 5 comment A quantity measuring weak non-compactness Certainly not! The weak$^*$ topology on $Y^{**}$, relativized to $Y$, is the weak topology on $Y$. Apply this to $Y=\ell_\infty$. That $\ell_\infty$ is a dual space is irrelevant. Jan 5 comment A quantity measuring weak non-compactness Wrong, Dongyang. You are confused by the two ways of embedding $X$ into $X^{(4)}$ I mentioned in my answer. Just think about $\ell_\infty$ as being a space that contains $c_0$ rather than as being $c_0^{**}$ and you'll see. In general, if $X$ is a closed subspace of $Y$, then the intersection of the weak$^*$ closure of $X$ in $Y^{**}$ with $Y$ is just $X$ itself. Jan 4 comment A quantity measuring weak non-compactness You ask some interesting questions, Dongyang, but rarely give any indication of why you are interested in the question or what information you have about it. You would probably get more "action" if you would take the time to give this information. Jan 4 answered A quantity measuring weak non-compactness Dec 27 comment Is $\ell_p$ $(10$. Dec 14 comment Embedding of real trees into $\ell_1(\Gamma)$ @Mikhail Ostrovskii: Do you know of any one dimensional metric space that does not embed into an RNP Banach space? Dec 14 comment Embedding of real trees into $\ell_1(\Gamma)$ @YCor Yes, that is the paper. I don't have the paper with me here. It is where we describe the constructive procedure for building a general $\ell_1$-tree. We do not state it as result that real trees embed isometrically into $\ell_1$ spaces, which we assumed was well known. Dec 14 comment Embedding of real trees into $\ell_1(\Gamma)$ I guess you haven't you read my paper with Lindenstrauss, Preiss, and Schechtman on $\ell_1$-trees. Embed any interval or half line of the tree into a one dimensional space. Take any branch point and add a new $\ell_1$ dimension and move an interval or ray branching off in this new direction. Continue transfinitely. Dec 13 awarded Yearling Dec 12 awarded Good Answer Dec 7 revised A question on weakly $p$-convergent sequences Corrected summable to convergent. Dec 7 comment example of an $\ell_1$-saturated Banach space without an unconditional basis Take the $\ell_1$ sum of any sequence of finite dimensional spaces whose GL constants tend to infinity. Dec 6 comment Existence or construction of a sequence of orthogonal matrices with three properties Use Walsh matrices. Dec 2 comment Banach spaces $X$ with $\ell_2(X)$ not isomorphic to $L_2([0,1],X)$ @NateEldridge: Tomek was using standard Banach space theory notation, which, as you point out, conflicts with what is used e.g. in geometric group theory.