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There are no really simple proofs of the characterization of finite dimensional $1$-injective spaces, but there are various proofs of even generalizations that do not directly rely on the Kelley-Goodn …

Somehow I missed this post two years ago. In the referenced paper, we proved something stronger than the statement that the two dimensional subspaces of $\ell_\infty^3$ mentioned by the OP are not iso …

No. Google "hereditarily indecomposable Banach spaces" to see that there exist separable Banach spaces in which all complemented subspaces are either of finite dimension or of finite codimension. Some …

$$X \cong \ell_p(X) \cong \ell_p(Y \oplus E) \cong \ell_p(Y) \oplus \ell_p(E) \cong Y \oplus \ell_p(Y) \oplus \ell_p(E) \cong Y \oplus \ell_p(X) \cong Y \oplus X$$. Only the third isomorphism requi …

The answer is negative. Take a biorthogonal sequence $(x_n,x_n^*)$ in $E$ such that $(x_n)$ converges to some unit norm vector $x_0$. Set $T=\sum_n 2^{-n} \|x_n\|^{-1} x_n^* \otimes x_n$. Such a seque …

The answer is positive for $\kappa = \omega$. The space $X$ is the $\ell_1$ sum of a sequence $(E_n)$ of finite dimensional spaces such that for every $\epsilon>0$ and every finite dimensional $E$, $E …

No, because the Schauder basis for $C[0,1]$ is perfectly reproducible. See the second paper by Lindenstrauss and Pełczyński, Contributions to the Theory of the Classical Banach Spaces, J. Functional …

As Jochen commented, you need $X$ to be reflexive, and this is sufficient. It is enough to show that the unit ball of $L_\infty(X)$ is closed in the reflexive space $L_2(X)$. But the injection from $ …

In my weak$^*$ basic sequences paper with Rosenthal we proved that if $\ell_1$ embeds into $X^*$ and $X$ is separable, then $c_0$ is isomorphic to a quotient of $X$.

Question 2 has a negative answer. Hint: Show that if question 2 has a positive answer, then every Hilbert-Schmidt operator must be of trace class.

Probably Spyros has in mind something like the following. Suppose you have a semi-normalized basic sequence $(x_n)$ in $V$ with biorthogonal functionals $(f_n)_n$ in $V_0^\perp \subset V^*$. Take any …

Gro-Tsen's answer to your previous question provides a counterexample if you define $D$ to be all vectors in $\ell_2$ that are of the form $\sum_n a_n f_n$, where $f_n = e_n + e_{n+1}$, $(e_n)$ is the …

IIRC Figiel and I reasoned this way: The dual norm to Tsrilson's original norm has the property that for any vector $x$ in $c_{00}$, $$ \|x\| \ge \|x\|_{c_0} \vee (1/2) \sup \sum_{k=1}^n \|E_k x\|, $$ …

Ozawa, Schechtman, and I finally wrote up what we know on this question. The estimate is that for every $\epsilon > 0$ there is a constant $C_\epsilon$ so that for every $n$, $\lambda(n)\le C_\epsilon …

No. $(\sum_{n=1}^\infty \ell_2^n)_{c_0}$ is a quotient of $c_0$. As I say to my students, "You know this; the problem is to figure out why you know this". :)