Questions tagged [compactness]

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Compact embedding of anisotropic Sobolev space

I am wondering if the embedding from $W^{2,1}_p(\Omega \times [0,T])$ to $C^{\alpha,\alpha/2}(\Omega \times [0,T])$ is compact, for some suitable domain, $p$ and $\alpha$. I have found some results. I ...
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  • 178
8 votes
3 answers
455 views

Structure theorems for compact sets of rationals

Everyone knows the Heine-Borel theorem characterizing compact subsets of Euclidean space. For any $n \in \mathbb N$ a set $A \subseteq \mathbb R^n$ is compact just in case it is closed and bounded (in ...
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35 views

An algebraic analogue of the Bolzano-Weierstrass for formal limits?

The Bolzano-Weierstrass Theorem is a very useful result in the theory of metric spaces. It states that given a compact space $X$, a sequence $(u_n) \in X^\omega$ always has a subsequence $(u_{n_k})\in ...
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1 vote
0 answers
71 views

Closure of finite rank operators on $L^p$

It well-known that, an operator $T:H\to H$ on a Hilbert space, is compact if and only if T is limit of finite rank operators. Besides this, the results by Per Enflo 1973 shows that this results is ...
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0 votes
1 answer
36 views

Exhaustions of product subsets by smaller product subsets

Let $X$ be a compact metric space, $A,B\subset X$ be subsets and $f\colon X\times X\to \mathbb{R}$ a continuous function that is strictly positive on $A\times B$. Do there exist increasing sequences ...
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1 vote
1 answer
127 views

Is the restriction of a projection to a compact subset a quotient map?

Let $(X, \mathcal{T}_X)$ and $(Y, \mathcal{T}_Y)$ be topological spaces, $Z = X \times Y$, $\mathcal{T}_Z$ be the product topology on $Z$, $f : Z \to X$ be defined by $f(x, y) = x$, and $C \subset Z$ ...
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  • 387
0 votes
1 answer
91 views

Is the Čech–Stone compactification of the integers always a retract of an extremally disconnected space?

Probably $\beta \mathbb N$ is not an absolute retract (is there an easy argument for this?), but I'd be interested to know what happens in the class of extremally disconnected (compact) spaces. Is it ...
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5 votes
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228 views

Is this property of continuous maps equivalent to some more familiar condition?

Let $f : X \rightarrow Y$ be a continuous map. Suppose that, for each collection of open sets $\{ V_i \}_{i \in I}\subset X $, $$ \bigcup_{U \subset Y \text{ open}, \ f^{-1}(U) \subset \bigcup_{i \in ...
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  • 4,283
4 votes
2 answers
305 views

Arzelà-Ascoli for $C_b(0,1)$? Or more generally, why is that continuous functions "live most naturally" on compact spaces?

I’m wondering if there is a version of Arzelà-Ascoli for continuous functions on not-necessarily compact metric/Hausdorff spaces $X$, i.e. a characterization of the compact subsets of $C_b(X)$ (under ...
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  • 211
7 votes
2 answers
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Is the union of a compact and the relatively compact components of its complementary in a manifold compact?

I was thinking of a way to prove this and I realised that for my approach the lemma from the title would be useful, and it´s an interesting question on its own. Obviously it is true if the manifold is ...
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6 votes
1 answer
629 views

Is a closed subset of an extremally disconnected set again extremally disconnected?

Let $T$ be a compact Hausdorff extremally disconnected set (so $T$ is a compact Hausdorff space, such that the closure of each open subset is again open). Let $S \subseteq T$ be a closed subset. ...
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-1 votes
1 answer
108 views

Definition of a $\psi$-Banach space [closed]

Let $X$ be a Banach space. Let $\mathcal{F}$ be the family of all the bounded subsets of $X$. If $I$ is the identity map on $X$, we shall denote by $\operatorname{span}\{I\}$ the vector space ...
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2 votes
0 answers
111 views

Conditions replacing compactness

Reading this book, the authors used the following "classic" idea: Let $X$ be a Banach space and $C$ a nonempty, weakly compact, convex subset of $X$. Let $T: C \rightarrow C$ be a ...
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3 votes
0 answers
81 views

About the nilpotency of a subgroup

Let $G$ be a compact group. Let $\mathcal N$ be a family of closed normal subgroups of nilpotency class at most $k$. Assume that $\mathcal N$ is closed under finite intersections and $\bigcap_{N\in\...
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1 answer
45 views

A MNC with maximum property but not singular

Let $E$ be a Banach space, $\mathfrak{M}_E$ indicate the family of all nonempty bounded subset of $E$, $\mathfrak{N}_E$ the family of all relatively compact sets, and $Ker \mu=\{X\in \mathfrak{M}_E$ ...
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1 vote
0 answers
119 views

A question about Theorem 2.3.1 in Tate's thesis [closed]

I don't understand how to prove a conclusion in the Theorem. When k is $p$-adic, the subgroups 1+$p^{v}$, $v>0$, of $u$ $(|u|=1)$ form a fundamental system of neighborhoods of $1$ in $u$, We must ...
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4 votes
1 answer
127 views

Can the degree of $k$-nilpotence of a simple simply connected compact Lie group be in $(0,1)$?

Let $G$ be a simple (i.e. every proper normal subgroup is discrete) simply connected compact Lie group. Define the degree of $k$-nilpotence of $G$ to be the Haar measure of the set $$\{(x_1,\dotsc,x_{...
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1 vote
1 answer
61 views

Does the compactness of parameter of distribution function imply the compactness of the distribution (or probability measure) in Wasserstein space?

For a family of probability measures sharing the same form of distribution function $F(x; p)$ with different parameters (i.e., $p$'s), if the parameter falls in a compact subset of real line, can we ...
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1 vote
1 answer
93 views

Limit of $u^\epsilon_t + u^\epsilon_x - \epsilon u^\epsilon_{xx} + \epsilon u^\epsilon_{xxx} = 0$ as $\epsilon \to 0$

Consider the initial-value problem associated to the PDE $u^\epsilon_t + u^\epsilon_x - \epsilon u^\epsilon_{xx} + \epsilon u^\epsilon_{xxx} = 0$. To prove that, as $\epsilon \to 0$, the weak solution ...
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0 votes
2 answers
274 views

A question about locally compact spaces

Recently I read a book about linear algebraic group written by Ian Macdonald. There is a conclusion which I can't prove. It says that if $X$ is locally compact Hausdorff space, then $X$ is compact if ...
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3 votes
1 answer
168 views

Can Tychonoffs theorem for a countable number of spaces be proven with ZF plus the axiom of (countable) dependent choice?

It can be proven without any form of infinite choice that the product of two compact spaces (and thus any finite product) is compact, while on the other hand, it is well known that the general form of ...
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  • 1,015
5 votes
1 answer
135 views

Compactness of symmetric power of a compact space

Suppose I have a compact metric space $(X,d)$ and let $\mathcal{X}=X^K$ be the product space. Consider the equivalence relation $\sim$ on $\mathcal{X}$ given as: for $\alpha,\beta\in \mathcal{X}$, $\...
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0 votes
1 answer
127 views

Essential spectrum under perturbation

Given a Banach space $X$ and a bounded linear operator $T$ on $X$. It's well known that the essential spectrum of $T$ is invariant under additive compact perturbation. My question is about minimal ...
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6 votes
2 answers
602 views

Is every continuous action of a compact topological group closed?

I am reading Bredon's Introduction to compact transformation groups, and came across the following result and proof on page 34: Even though he writes "Recall our standing assumption that $X$ is ...
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1 vote
0 answers
182 views

Is the set of compact operators closed with the strong topology?

It is well-known that the space of compact operators over Banach spaces is closed within the norm topology. My question: Let $X$ be a Banach space. Considering the strong topology (defined by ...
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4 votes
0 answers
122 views

Double commutant of compact operators

So my question is straightforward. Let $\mathfrak{X}$ be a (complex, if necessary) Banach space and $K\colon\mathfrak{X}\to\mathfrak{X}$ a nonzero compact operator. Denote by $\mathcal{C}(K)$ the ...
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5 votes
1 answer
345 views

Who are the owners of the compactness theorem in $L^p(\Bbb R^d)$?

As the title says, I am interested to know Who are the owners of the compactness theorem in $L^p(\Bbb R^d)$. There is some confusion in the literature. Let recall that the compactness theorem in $L^p(\...
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  • 720
1 vote
1 answer
110 views

Density and compactness of Boolean embeddings

Let A and B be Boolean algebras and $h:A\rightarrow B$ a Boolean embedding. If every element of $B$ can be expressed both as a join of meets and as a meet of joins of elements in $h(A)$, then the ...
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1 vote
0 answers
129 views

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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  • 111
2 votes
1 answer
138 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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  • 1,457
1 vote
1 answer
131 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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4 votes
0 answers
100 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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7 votes
1 answer
205 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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  • 1,547
0 votes
1 answer
123 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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  • 627
2 votes
0 answers
322 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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  • 627
1 vote
1 answer
200 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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  • 2,058
3 votes
2 answers
191 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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2 votes
1 answer
145 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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2 votes
1 answer
120 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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  • 301
1 vote
1 answer
45 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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  • 921
4 votes
2 answers
338 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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  • 3,594
8 votes
1 answer
156 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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1 vote
0 answers
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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1 vote
1 answer
98 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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7 votes
1 answer
211 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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4 votes
2 answers
233 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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2 votes
1 answer
292 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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  • 627
5 votes
1 answer
519 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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  • 307
0 votes
0 answers
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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  • 223
5 votes
1 answer
166 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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