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The OP's intuition that $X:=(\oplus \ell^2_{p_n})_2$ with $p_n$ tending to $1$ (and $1<p_n < \infty$) provides an example is correct. Indeed, $X$, being an $\ell_2$ sum of strictly convex spaces, is …

Yes. If $R:\ell_q \to X$ and $TR$ is not compact then there is a normalized block basis $u_n$ of the unit vector basis for $\ell_q$ s.t. $TRu_n$ is bounded away from zero. Let $V:\ell_p \to \ell_q$ …

Yes. Choose $n_1<n_2<\dots$ s.t. for all $m$ $$ \sup_{x^{*}\in B_{X^{*}}}\sum_{k=n_m}^\infty |\langle x^*, x_k \rangle|^p < \epsilon^p/4^m. $$ Set $\lambda_k = 1$ for $k<n_1$ and $\lambda_k = 2^{-k} …

No. Consider the identity operator on $\ell_r(S)$ with $p<r<\infty$ and $S$ uncountable. Or consider any weakly compact operator with non separable range into a $C(K)$ space or an $L_1$ space and us …

No, because $L_r$ has cotype 2 for $r\le 2$. In fact, every bounded linear operator from $\ell_q$ into $L_r$ is compact when $r \le 2 < q$.

No. Consider $z_{2n+1} - z_{2n}$.

The equivalence of the two definitions of $\mathscr{L}_p$-space involves an elementary perturbation argument that is probably contained in the original article of Lindenstrauss and Pelczynski. That k …

Suppose that $A$ is a bounded subset of $X$ and $X$ is a subspace of $Y$. Then \begin{equation} wk_Y(A) \le wk_X(A) \le 2 wk_Y(A),\ \ \ (\#) \end{equation} and $2$ is the best constant in the right …

In the theorem you quote, part of the definition of subsymmetric includes the hypothesis that the sequence is unconditionally basic. If you do not include this in the definition of subsymmetric, then …

Even if $K$ is just weakly compact, $X_K$ is isometrically isomorphic to a dual space with the weak$^*$ topology agreeing with the weak topology from $X$ on $K$. You can get a weakly compact convex s …

The answer is yes by a small perturbation argument. For a small $\epsilon$, enlarge $F$ so that $F$ $1+\epsilon$-norms $E$. Get $T$ from the condition. Choose any Auerbach basis $u_k$ for $E$ …

Given a linear operator $A$ from a Banach space $E$ to a Banach space $F$, the quantity $\gamma_2(A)$ is the infimum (actually, it is a minimum) of $\|X \| \cdot \|Y\|$, where the minimum is over all …

Not unless $p=1$, in which case you interpret $\ell_q$ to be $c_0$. The most interesting case is $p=2$. Tomczak's theorem says that you can compute $\pi_2(T)$ for $T: \ell_2^n \to X$ up to the con …

In every Banach space that has a Schauder basis there is a Schauder basis that is not unconditional (meaning that there is a rearrangement of the basis that is not a basis). See Pełczyński, A.; Sing …

$C^k(M)$ is isomorphic to $C(M)$ and hence, by Milutin's Theorem, to $C[0,1]$. The first statement is more or less obvious since the norm on $C^k(M)$ is equivalent on a $k$-codimensional subspace to …