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The separable spaces that do not have your property for bounded $(x_n)$ (namely, there is a sequence in the unit sphere satisfying your limit condition) are characterised in the paper ``Thickness of t …

Probably the authors refer to the space $L_0(\mu)$ of all (equivalence classes) of measurable functions. This is a complete metric space in the metric I have mentioned in a comment above or, which is …

Your norm, call it $N$ for \TeX-nical simplicity, is not WLUR. You want to know whether $N(f)=1$, $N(f_n)\to1$, $N(f+f_n)\to 2$ imply that $f_n\to f$ weakly. For a counterexample let $f$ be the const …

The space $C[0,1]$ has the Daugavet property; in particular, for a finite-rank projection $\|I-P\|=1+\|P\|$, which equals $2$ if $\|P\|=1$.

As for Question 3, note that an absolutely summing operator is completely continuous, i.e., maps weakly null sequences to null sequences, a property not shared by $i_{p,q}$ if $p>1$. Therefore this qu …

Yes, it does. This is essentially an unpublished result due to Hermann Pfitzner, see III.3.6 and III.3.7 in $M$-ideals in Banach spaces and Banach algebras by P. Harmand, W. Werner and myself (Zbl 078 …

If $X$ is isometric to a space $L_1(m)$, then $X^{**}$ is isometric to a (highly nonseparable) $L_1$-space over some measure space $(\Omega, \Sigma, \mu)$, by the duality of abstract $L$- and $M$-spac …

The norm of (even a continuous convex function on) a separable Banach space is Gâteaux differentiable on a dense $G_\delta$-set (Mazur), but the canonical norm on the nonseparable Banach space $\ell_1 …

Consider $\delta_{1/n}-\delta_0$; this defines a weak$^*$ null sequences which is not weakly null; e.g., $\langle \delta_{1/n}-\delta_0, \chi_{\{0\}} \rangle \not\to 0$. So if $\Omega$ contains a nont …

The key here is the isometric embedding of $K(E,F)$ into the space of continuous functions on the compact space $M=B_{E^{**}}\times B_{F^*}$. Suppose that $A$ is WOT$^*$ sequentially compact; $A$ is b …

I find the following criterion useful: A sequence $(x_n)$ is Cauchy iff for all subsequences $(x_{n_{k+1}}-x_{n_k})$ tends to $0$. This works for the norm topology, the weak topology and the weak$^*$ …

On page 148 of Convexity Theory and its Applications in Functional Analysis, L. Asimow and A.J. Ellis say that every Dirichlet algebra (an algebra where the real parts of its elements are dense in the …

The answer is no. Bourgain and Pisier have given a counterexample (A construction of $\mathcal{L}_\infty$-spaces and related Banach spaces. Bol. Soc. Bras. Mat. 14, No. 2, 109-123 (1983). See Zbl 0586 …

It is known that $C(K)$, for infinite $K$, contains a copy of $c_0$, hence it does not have nontrivial type (meaning $>1$) or nontrivial cotype (meaning $<\infty$).

Let's consider the following situation: $E$ and $F$ are Banach spaces, $D\subset E$ is a dense subspace, $Q: E \to F$ is a continuous linear operator, and its restriction $Q_0$ to $D$ is a quotient ma …