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The converse is indeed true; in particular $C_0(K)$ does not have nontrivial $M$-summands if $K$ is connected. The argument to show this in [P. Harmand, yours truly and W. Werner, $M$-Ideals in Banach …

If $c_0$ were isometrically isomorphic to $c_0\oplus_1 c_0$, the dual $\ell_1$ would be isometrically isomorphic to $\ell_1\oplus_\infty \ell_1$. Such an isomorphism, $A$, would map extreme points of …

Note that $\mathrm{diam}( \overline {\mathrm{conv}}(C) ) = \mathrm{diam}(C)$, and note that $\|e_i-e_j\|_{p,\infty}=1$ for $i\neq j$. This proves your inequality $\|x-y\|_{p,\infty}\le 1$. However, …

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 …

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 …

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 $ …

I think $\mathcal D$ is dense: Let $f\in L^2([0,1],\mathbb{R}^n)$, $\varphi\in\mathbb{R}^m$ and $\delta>0$. Take $g\in C^1([0,1],\mathbb{R}^n)$ such that $\|f-g\|_{L^2}<\delta$. Let $\psi=C\varphi+Gg( …

If I understand your question correctly, you are asking if $k'\le ck$ for a constant $c\ge1$ implies that $\mathcal{H}_{k'}\subset \mathcal{H}_{k}$. Now Theorem 3.11 in V.I. Paulsen, M. Raghupathi, An …

Actually, a somewhat weaker condition is sufficient, namely that $x_0$ is a strong extreme point, meaning that $\|z_n\|\to0$ whenever $\|x_0\pm z_n\|\to1$. This is the same as saying that for each $\v …

One can look at Pietsch's Eigenvalues and s-numbers or König's Eigenvalue Distribution of Compact Operators to find that such an operator is 2-summing and hence Hilbert-Schmidt, and its eigenvalues sa …

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 $<\infty$).

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 …

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

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$^*$ …