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While digging through old piles of notes and jottings, I came across a question I'd looked at several years ago. While I was able to get partial answers, it seemed even then that the answer should be …

If you only require isomorphism in the sense of an invertible, continuous linear bijection, then the answer is yes. If you require isometric linear isomorphism, the answer is no (because the unique is …

Choose finite sets $A_n\subset{\bf N}$ with $\max(A_n)<\min(A_{n+1})$, and then define $T:c_{00}\to c_0$ by $T(e_n)=\chi_{A_n}$. It is easy to check that $T$ extends to a norm 1 linear map $\ell^2\to …

I'm not a Banach-space specialist, and don't have a copy of Lindenstrauss-Tzafriri to hand (though maybe the Albiac-Kalton book would be a friendlier read, if your library has a copy?) but it seems to …

Contrary to my original muddled guess, the answer is no: the problem is that your `extension operators' don't give enough control over what happens in the gap between $U_\infty$ and $U_r$. For a conc …

Does this paper of Kalton do the trick? (disclaimer: I haven't read through the details)

(Posting this comment as an answer, just so the question doesn't still show up as "unanswered".) Your first claim is not justified (as you yourself suspected). If $T:X\to Y$ has separable range, th …

No. Take $X=\ell_\infty({\bf N})$ and take $E_n = \operatorname{span}(e_1,\dots, e_n)$. Then $Y=c_0({\bf N})$ which is well-known – by a non-trivial argument – to be uncomplemented in $X$ (in the sens …

The answer to Q2 is negative. One way to see this (which might not be the first, and might not be the easiest) is to combine the following results. A separable dual (Banach) space has the Radon-Niko …

Examples were constructed (about two years ago?) by Argyros and Haydon. See this blog post for some non-technical discussion. It seems worth noting, as one is almost obliged to, that the space origina …

... are all isomorphic to $l^1$ on some other index set. At least, that much I "know" from 2nd-hand sources, since the original proof is apparently in a paper of Köthe from the 1930s 1960s (in German) …

The first counterexample that comes to my mind, which is probably overkill (memory is easier than thought!) is J. Bourgain, A counterexample to a complementation problem. Compositio Mathematica, Volu …

I will take a roundabout way to defining this ideal, because (a) this route is how my collaborators and I came to it (b) this alternative definition, rather than the standard one, may suggest a direc …

For this question, all Banach spaces are over the reals. Let $1\leq p<\infty$. Recall that a sequence $(x_n)$ in a Banach space $E$ is weakly $p$-summable if $$ \Vert (x_n) \Vert_{p,w} := \sup_{\gamm …

The following questions are about the history of a particular result in functional analysis, hence not "mathematical questions" per se; but I think they are relevant to the business of writing researc …