Just an aside to @Nate's answer: the point masses are at pairwise distance 2.

It seems to me that ticking that box would just cascade the paper down within the same publishing house, but not elsewhere, e.g., not from Springer Journal A to Cambridge Univ. Press Journal B. The author may have had different ideas where the paper should end up.

Please note: "contains an isomorphic copy of $\ell_1$'' is not the same as "contains $\ell_1$ as a subset''.

Bill, no problem!

@Bill: I think I said so in my answer...

@Dieter: If the sequence is weak$^*$ convergent to $0$ and weakly convergent to something, then something $=0$; so one just has to check weak convergence to $0$.

@David Tonhofer: Actually, here Nachlass (legacy) refers to papers the deceased person wrote during his lifetime and were found after his death. It is not that somebody else writes the Nachlass, but it may be edited, of course.

@dohmatob You will find results like this in the books that I have mentioned!

Since $I_K$ is a Hilbert-Schmidt operator, the eigenvalue sequence is in $\ell_2$. -- A detailed analysis of the eigenvalue distribution of compact (and other) operators on Banach spaces can be found in the monographs by H. K\"onig (Eigenvalue distribution of Compact Operators) and A. Pietsch ($s$-Numbers and Eigenvalues).

Even shorter, at least in print, is $c_p$ by C.A. McCarthy (Israel. J. Math. 5, 249--271 (1967)), about the Schatten classes..

This appears to be one instance of what Pietsch refers to by saying ``Mathematicians invent their own history.''

If I am not mistaken, the condition $f(\frac12x + \frac12y)\le \frac12 (f(x)+f(y))$ is sometimes called midpoint convexity.

@Giorgio: Yes, separability of the bidual is good enough, e.g., the bidual of the James space is separable. And a separable Grothendieck space is reflexive. -- Q2 is true in a Grothendieck space (or any other space with a wsc dual); but I read the question as about general Banach spaces.

Call the condition in Q1 Cauchy Grothendieck. Let $X$ have this property. If $(x_n^*)$ is w$^*$ null, it has a limit $x^{***}\in X^{***}$. To show that it is $0$, consider the w$^*$ null sequence $(x_1^*, 0, x_2^*, 0, \dots)$ interlacing the given sequence with $0$. It has a limit $y^{***}\in X^{***}$ since the space is Cauchy Grothendieck. Now along the odd integers, the new sequence tends tends to $x^{***}$, along the even integers it tends to $0$. Hence $x^{***}=0$.