# Questions tagged [compactness]

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106
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### Relative compactness… but what is the toplogy?

The following Theorem was described in a text I was reading as a compactness result. The proof is probably too advanced for me but I was just wondering with respect to what topology we have ...

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99 views

### Weakly convergent sequence and compensated compactness

This question is about a claim made in the proof of theorem 2.1.1 in the book Hyperbolic Conservation Laws and the Compensated Compactness Method by Yunguang Lu. (For simplicity I will only write done ...

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111 views

### Gradient-like dynamical systems

I've tried asking this question on Mathematics site, but I only got an upvote and no answer. I've searched online, tried to find something about this topic, but I haven't found much (and the things I ...

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86 views

### Examples of indefinite Einstein non-Ricci-flat metric on solvmanifolds

A solvmanifold (resp. nilmanifold) is a compact quotient of a solvable (risp. nilpotent) Lie group $G$. Usually one consider a co-compact lactice $\Gamma$ on $G$ and the solvmanifold is $G/\Gamma$. ...

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182 views

### Characterization of pretty compact spaces

This is a cross post from MSE.
I believe that the following problem have already been considered by some sophisticated topologist.
Definition 1. A non-compact Hausdorff topological space $X$ is called ...

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84 views

### Tightness on a set $A$ implies tightness on a set $B$ where $A\subset B$?

From the book Billingsley - Convergence of probability measures, 1999, we have the following definitions of tightness and relative compactness and the Prohorov's theorem:
Tightness: Let $\Pi$ be a ...

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83 views

### Compact embedding of the Sobolev space $H^m(\Omega)$ and $L^2(\Omega)$ from Rellich-Kondrachov theorem

From the Rellich-Kondrachov theorem we know that $H^m(\Omega)\hookrightarrow_c L^2(\Omega)$ when $\Omega$ is bounded of class $C^1$ and $m\geq 1$ is an integer. Also this is not true if $\Omega:=\...

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119 views

### $L^2$ uniform integrability in terms of Fourier coefficients

Given a bounded sequence $(f_n)_n$ in $L^2(\mathbf{T})$ where $\mathbf{T}:=\mathbf{R}/\mathbf{Z}$, the strong compactness of $(f_n)_n$ is equivalent to $$\lim_N \sup_n \sum_{|k|\geq N} |c_k(f_n)|^2=0,$...

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144 views

### Relative compactness in topological spaces (reference request)

Motivation and context: For a subset $S$ of a metric space $(M,d)$, the following are two very classical compactness results in Analysis:
1a) The set $S$ is compact if and only if each sequence in $S$...

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107 views

### Space of embedded minimal surfaces of fixed genus in a generic $3$-manifold

Let $M^3$ be a closed, connected and oriented smooth $3$-manifold, and fix an integer $g \geq 1$. Is it true that for a generic set of Riemannian metrics on $M$ the set of closed, connected and ...

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103 views

### Compactness of a sequence

Let $E$ be a Banach space, $T:E\rightarrow E$ a continuous, norm-bounded, and nonlinear mapping., and $\{x_n\}_{n\in\mathbb N}$ such that $$x_{n+1}=T(x_n),\:\forall n\in \mathbb{N}:=\{0,1,\cdots\}.$$
...

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40 views

### S-unital compact rings are profinite

It is well-known that compact Hausdorff topological unital rings are profinite. The proof generalises to (left or right) s-unital rings (i.e. rings such that for all $r\in R$ we have $r\in Rr$ or for ...

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250 views

### How to describe the compact real forms of the exceptional Lie groups as matrix groups?

I know that $G_2$ can be described as the subgroup of $SO(7)$ preserving a specific element of $\Lambda^3(\mathbb{R}^7)^*$. It can thus be realized as a matrix group. Prof. Robert Bryant did describe ...

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100 views

### Subspaces of $L_p([0,1])$ whose unit ball is compact for the topology of convergence in measure

Any information about the following questions would be welcome.
I wonder whether there are (well-known or easy) closed and infinite dimensional subspaces of $L_p([0,1])$ ($1<p<\infty$) whose ...

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51 views

### $ \{x\in X:h(x)\leq r\} $ is sequentially compact subset of $X$?

Let $(X,\|.\|)$ be a reflexive Banach space and $(D,\|.\|)$ be a Suslin subspace of $X$ such that $D$ is weakly closed subset of $X$.
Take $h:X\to [0,+\infty]$ such that $h(x)=\|x\|$ if $x\in D$ and ...

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91 views

### $C$ is **weakly compact** or **weakly locally compact**?

Let $X$ be a separable Banach space such that $X$ and its dual $X^*$ have Radon-Nikodym property. Let $C$ be a convex, closed and bounded subset of $X$.
Can we say that $C$ is weakly compact or ...

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168 views

### Does a compact Lie group have finitely many conjugacy classes of maximal Abelian Lie subgroups?

Let $G$ be a compact Lie group. An Abelian Lie subgroup $A \leq G$ is a maximal Abelian Lie subgroup if, for any Abelian Lie subgroup $A'$ such that $A \leq A' \leq G$, then $A' = A$.
Of course any ...

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224 views

### Compact spaces whose compactness does not come from a product of compact spaces

For the (Hausdorff) compact spaces I can think of, compactness is established either using a product of compact spaces (including the Heine-Borel Theorem, the Banach-Alaoglu Theorem, Stone-Čech ...

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148 views

### Compact embedding of space of signed Radon measures into Sobolev space $W^{-1,q}$ from Evans paper; Does it work in one space dimension?

Background: I work on a PDE problem where I have some approximating sequence of measure-valued functions and I need to compactly embed it into some negative Sobolev space $W^{-m,q}$ on the bounded ...

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336 views

### Existence of injective compact operators

We know that if $X$ is a separable Banach space, then for every infinite dimensional Banach space $Y$, there exists an injective compact operator from $X$ to $Y$.
My query is for every Banach ...

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126 views

### Spectral Theorem for compact self-adjoint operators on real Hilbert spaces [duplicate]

Is the spectral theorem for self-adjoint compact operators on a Hilbert space also true if the Hilbert space is real (instead of complex)?
Wikipedia says this is true.
However, it seems to me that ...

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141 views

### Reeb stability counterexample: foliation in $S^{n-2}\times S^1\times S^1$ with non-diffeomorphic leaves

Reeb's global stability theorem requires the foliation to be of codimension 1. As a counterexample, in "Geometric theory of foliations", Camacho and Lins Neto present the following.
Consider the ...

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381 views

### Is the separability of the space needed in the proof of the Prohorov's theorem?

The Section 5 of the book:
Billingsley, P., Convergence of Probability Measures, 1999,
studies Prohorov's theorem. A short reminder is given below.
Let $\Pi$ be a family of probability measures on ...

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239 views

### Can I get away without using Arzela-Ascoli?

I am currently thinking of function-valued random variables. In order to prove a result, I need to approximate by (function-valued) step functions. This naturally leads to the idea of chopping up the ...

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507 views

### On weak compactness of the unit ball in a reflexive Banach space

It is a well known result in functional analysis that a Banach space $X$ is reflexive if and only if the unit ball is weakly compact (compact in the weak topology). This result is also known as ...

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129 views

### About the finished, $\aleph_0$…-compactness

Definitions :
$(E,d)$ a metric space is finished-compact if any covering of $E$ by open, we can extract a finite subcover
$(E,d)$ is $\aleph_0$-compact if for any infinite covering of $E$ by open, we ...

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121 views

### A non-condensing operator with a power condensing

Let $\alpha$ to be the Kuratowski measure of non-compactness, in a Banach space $E$.
It's very easy to prove that $\alpha (D_1\times D_2)\leq \alpha (D_1)+\alpha (D_2)$, where $D_1$ and $D_2$ are ...

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99 views

### Subspaces of compact spaces and quotients of Hausdorff spaces

Let $\operatorname{Top}$ be the class of topological spaces. Furthermore, let $\mathcal{U}\subset\operatorname{Top}$ and $\mathcal{V}\subset\operatorname{Top}$ classes satisfying the following ...

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662 views

### CG spaces from the perspective of sheaves over compact Hausdorff spaces

A compactly generated space is a space $X$ such that $f : X \rightarrow Y$ is continuous if and only if $K \rightarrow X \stackrel{f}{\rightarrow} Y$ is continuous for each compact hausdorff space $K$....

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150 views

### Bounded ball measure on compact metric space

Fix $c>1$. Let $(X,d)$ be a separable compact metric space, does there necessarily exist a Borel probability measure $\nu$ on $(X,d)$ such that
$\operatorname{sup}_{x \in X,r>0}\frac{\nu(\...

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69 views

### Finite sub-covering of ball in Hilbert space

Let $H$ be a separable Hilbert space, $B$ the closed unit ball in $H$, and $(x_n)$ be a dense sequence in $B$, $(e_n)$ be an orthonormal basis, and $\varepsilon>0$. Then $B$ is covered by the (...

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92 views

### Compactness of a special kind of Integral operators

Let $(S(t))_{t>0}$ be a continuous operator from $L^2(0,1)$ to its self and Let $K$ be the operator $$\eqalign{
& K:{L^2}(0,1) \to {L^2}(0,1) \cr
& f: \to (Kf)(x) = \int\limits_0^1 {k(...

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323 views

### Compact quasi-coherent sheaves

Let $X$ be a scheme. What are the compact objects in the category of quasi-coherent $\mathcal{O}_X$-modules? All references seem to discuss the derived category but I need the abelian category.

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412 views

### Compactness of set of indicator functions

Let $\chi_A(x)$ denote an indicator function on $A\subset [0,1]$. Consider the set
$$K=\{\chi_A(x): \text{ A is Lebesgue measurable in }[0,1]\}.$$
Is this set compact in $L^\infty(0,1)$ with respect ...

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120 views

### Compact operator

Let $k:[0,1]^2 \to [0,1]$ be a measurable function. Define $K:L^2([0,1])\to L^2([0,1])$ to be the operator:
$$
(Kf)(x) = \int_0^1\int_0^1 f(z) k(x,y) \mathbf{1}_{x\leq z\leq y} \ \mathrm{d}z \mathrm{d}...

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51 views

### Continuous injection of metric ball into Euclidean ball

This is a follow-up to this post.
Suppose that $(X,d_X)$ is a compact metric space with (finite) metric-capacity, defined by
$$
\kappa_X(\epsilon)\triangleq\sup\left\{
k : \exists x_0,\dots,x_k \...

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108 views

### Approximation argument in geometric flows

I'm studying by myself Mean Curvature Flow and I'm reading the paper "Interior estimates for hypersurfaces moving by mean curvature" by Klaus Ecker and Gerhard Huisken and I'm stuck in the following ...

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356 views

### When Stone–Čech compactification is totally disconnected

A topological space $X$ is totally disconnected if the connected components in $X$ are the one-point sets, and a topological space, $X$ is called completely regular exactly in case points can be ...

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222 views

### Uniqueness of limits and compactness implies closure

It is not difficult to prove that in a Hausdorff topological space every compact set is closed, and almost trivial that if in a topological space X every compact set is closed then X is T1. As ...

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83 views

### Approximation of deterministic problems in the PDEs with the stochastic ones

A lot of problems in PDE theory are solved in the following way:
The original problem is quite hard and we can't solve it, so we make the approximation problem that we can solve. Than we go back and ...

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148 views

### “Compactness in Measure” in Function Spaces

In Chapter 4.9 of the book "Measures of Noncompactness and Condensing Operators" (Vol. 55 of Operator Theory: Advances and Applications), the authors mention the property "compactness ...

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193 views

### Compact images of nowhere dense closed convex sets in a Hilbert space

Let $B=-B$ be a nowhere dense bounded closed convex set in the Hilbert space $\ell_2$ such that the linear hull of $B$ is dense in $\ell_2$.
Question. Is there a non-compact linear bounded operator ...

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514 views

### Weak*-convergence of signed measures

Let $X$ be a compact Hausdorff space and let $M(X)$ denote the space of signed measures that is naturally dual to $C(X)$, the space of continuous functions on $X$. I am interested whether the ...

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208 views

### A ZFC-example of a countably compact paratopological group which is not a topological group

Problem. Does there exist a ZFC-example of a countably compact Hausdorff paratopological group which is not a topological group?
(The problem posed 27 May 2015 by Alexander Ravsky on page 9 of Volume ...

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259 views

### Extending homeomorphisms between compact metric subsets

Let $X$ be a compact metric, second countable space with finite covering dimension. Let $A,B$ be two closed subsets of $X$. Assume that $h:A\to B$ is a homeomorphism.
Is it possible to extend $h$ to a ...

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155 views

### Compact embedding result

Let $\tau$ and $\ell$ be positive numbers. We know that the space $H^2(0,\ell)\cap H^1_0(0,\ell)$ is compactly embedded into $L^6(0,\ell)$. Now, is the space $L^2(0,\tau;H^2(0,\ell)\cap H^1_0(0,\ell))$...

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686 views

### Does point-wise weak convergence give weak convergence in $L^2(I;X)$?

Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly continuous on $X$, and $u_n$ are $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, ...

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47 views

### Does $K^{1/2} (t,s)$ inherit the continuity of $K(t,s)$?

Assume that $K(t,s)$ is a (1) symmetric, (2) continuous, and (3) positive definite kernel on $[0,1] \times [0,1]$. The spectral decomposition of $K(t,s)$ is:
$$
K (t,s) = \sum_{i=1}^\infty \lambda_i \...

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242 views

### Pullback of Morse form satisfies Palais Smale

Let $(\alpha,g)$ be a Morse-Smale pair on a closed smooth manifold $M$, i.e. $\alpha$ is a Morse form and $g$ a Riemannian metric on $M$ such that stable and unstable manifolds of the gradient vector ...

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89 views

### Compact generation of the category of D-modules on moduli stack of principal bundles for algebraic groups?

Let $k$ be an algebraically closed field of characteristic 0. Let $X$ be a connected smooth complete curve over $k$. Consider the moduli stack $\mathrm{Bun}_G$ of principal $G$-bundles on $X$ for ...