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Since weakly compact operators into $\ell_1$ are compact, and since by a result of Kadec and Pelczynski every non-weakly compact operator into $\ell_1$ fixes a copy of $\ell_1$, we have that if $X$ co …

Given an affine scheme $S=\operatorname {Spec (R)} $ and a closed subscheme $T=V(J)\subset S$, the restriction mapping $$\mathcal O(S)=R\to \mathcal O(T)=R/J$$ is obviously surjective. Applying this t …

EDIT: This isn't really an answer - see Andres' comment - but you might find it interesting, and it is too long for a comment. Your second question is vague, but has an affirmative answer in the foll …

Claim: The principle $A_{null}$ follows from continuum is RVM. Proof: Let $m$ be a total extension of Lebesgue measure. Suppose every vertical section of $A \subseteq \mathbb{R}^2$ is Lebesgue null. …

I don't know the answer. However, this is what I would try. $\alpha$ is constrained to have as a subword $ac^ma$, for nonzero $m$ and where I abuse notation and use $a$ to represent itself or its in …

If the $X_i$ are independent and uniformly distributed on $[0,1]$, the (finite) point process $\sum_1^n \delta_{nX_i}$ converges in law to a Poisson point process on $\mathbb R^+$ with unit intensity, …

You can assume less: $\pi$ faithfully flat, $F$ quasi-coherent, no assumptions on structure morphisms. Write down the functors and see immediately their how faithfulness and exactness depend on each o …

Let $C \subset S^1 \subset \mathbb{R}^2$ be a Cantor set. Let $H_C$ be its convex hull, the smallest closed subset of $\mathbb{R}^2$ containing $C$. Then $H_C$ is not a manifold with corners. Its boun …

See de Rham's "cutting corners" curve. Picture from an MO answer by Bill Thurston; also see a description there. The limiting curve is $C^1$ but not $C^2$. It has a tangent everywhere, but …

Federico already mentioned the keyword. The precise answer may be found among others as Theorem 13.16, of this book. (That theorem makes a restriction to real matrices, but that is not necessary).

Consider the following curve (very informally described): start from the origin in $\mathbb{R}^2$, then move from one unit up. Turn of an angle $\pi/4$ on the left and move from half of unit. Turn o …

Whatever almost complex submanifolds are, complex submanifolds should be a subclass. There are compact complex tori which admit complex sub-tori. One (natural) class of examples are given as follows: …

Upon reflection, I find this approach interesting enough, although the example you gave is too elementary to be sure. Provided your $K_G$ contains $\mathcal D_0:=C_c^\infty(0,\infty)$, your "bilatera …

A symmetric closed category is a closed category together with isomorphisms $$s:[A,[B,C]] \cong [B,[A,C]]$$ satisfying a few axioms: see Definition 1.1 of the paper ``On embedding closed categories" b …

In part 2 of Game Theory by Thomas Ferguson, example 2 'Turning Corners' on page 33, Thomas Ferguson mentions a so-called 'flipping-coin' game, where the Sprague-Grundy functions g(x, y) equals nim mu …