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Yes. Let $P$ be a (continuous linear) projection from $X_1 \times X_2$ (equipped with the sum-norm, say) onto $Y_1 \times Y_2$. For $x_1\in X_1$ consider $P(x_1,0)$, which can be written in the form $ …

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 $\|P\|=1$.

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 …

Only the subspaces mentioned by the OP are obviously closed. Let $V$ be a real vector space and $W\subset V$ a proper infinite-dimensional linear subspace. We shall endow $V$ with a norm so that $W$ w …

The question can be reformulates as follows. If $H$ is a Hilbert space, $X$ ($=B^∗$) is a Banach space, $T:H\to X$ is an operator, $i:X\to H$ is a continuous injection, can one estimate the $2$-summin …

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 suppose that $\|f\|=1$. Let $P$ denote the set of probability measures supported on the compact set $A$. Then $J(f)$ consists of all measures of the form $h\,d\mu$, $\mu\in P$, where, for all $x …