Search Results

Yes, if $X$ and $Y$ have property (C), then so does $X\oplus Y$ because property (C) is a three-space property; this is a result of Pol (Proposition 1 in the paper you are referring to). However, I a …

Absolutely not. Take the the $\ell_1$-sum of $\ell_\infty^n$ ($n\in \mathbb N$). If that embedded into $L_1(\mu)$, then you would have found $c_0$ in some ultrapower of $L_1(\mu)$, which is impossible …

Yes, they are the same. Pełczyński proved that strictly singular, strictly $\ell_1$-singular, strictly cosingular, and weakly compact operators on $L_1$ are all the same. This is Theorem 1 in A. P …

Yes, there is such characterisation. $C(K)$ contains no isomorphic copy of $\ell_1$ if and only if $K$ is scattered. Indeed, if $K$ is scattered then $C(K)^*$ is isometric to $\ell_1(K)$, so $C(K)$ ca …

According to the recent preprint by Koszmider, Shelah and Świętek under the generalised continuum hypothesis there is no such bound. In particular, one cannot prove the existence of such a bound worki …

Yes. This is Theorem 3.1 in L. P. Belluce, W. A. Kirk, and E. F. Steiner, Normal structure in Banach spaces. Pacific J. Math. 6, (1968), 433-440.

You cannot prove the statement about a.d. families from Q1 in ZFC only. For instance, it does not hold if $\mathfrak{c}=\omega_1$ and $2^{\omega_1} = \omega_2$. For details see J.E. Baumgartner, A …

I guess so. Here's an outline which reduces everything to the separable case. Let $p\in (1,2]$. Then of course $L_p[0,1]^I$ is finitely representable in $L_1$. Take an ultrafilter $U$ (on some ridic …

If you consider the spaces constructed by Gowers, Maurey and others exotic, then it turns out that the answer is also negative in the class of $C(K)$-spaces. Indeed, Koszmider constructed a compact, H …

Answer to Q1b is particularly easy as this space embeds into $c_0$, hence it is saturated with complemented copies of $c_0$. There is however a unified argument which works for all your spaces of int …

Yes, it can deduced from Lemma 1.2 combined with Proposition 2 of D. Alspach and Y. Benyamini, Primariness of spaces of continuous functions on ordinals. Israel J. Math. 27 (1977), 64–92. Anoth …

Jensen's diamond implies CH. See https://math.stackexchange.com/a/2073421/17929 https://en.wikipedia.org/wiki/Diamond_principle

Piotr Hajłasz' answer nails the problem, however, let me point out that there are easier examples of such pairs of spaces among spaces that have the same density. Suppose that $X$ fails to have a str …

Ben, $$\|x\|^\prime = \inf\Big\{ \big(\sum_{i=1}^n \|x_i\|^p\big)^{1/p}\colon \sum_{i=1}^n x_i = x, x_i\in X, n\in \mathbb N \Big\}\qquad (x\in X)$$ is the standard $p$-convex renorming. The hardish …

The Szlenk index is the answer. A space $C(K)$, where $K$ is infinite compact Hausdorff space, is embeddable into $c_0$ if and only if $K$ is homeomorphic to an ordinal below $\omega^\omega$ and if …