Questions tagged [compactness]

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Convolution of $L^2$ functions

Let $u\in L^2(\mathbb R^n)$: then $u\ast u$ is a bounded continuous function. Let me assume now that $u\ast u$ is compactly supported. Is there anything relevant that could be said on the support of $...
Bazin's user avatar
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6 votes
1 answer
139 views

When does base-change in topological spaces preserve quotient maps?

The question when $(-) \times X$ preserves colimits in topological spaces is well-studied. Since it always preserves arbitrary coproducts (disjoint unions), one only has to show when it preserves ...
Lennart Meier's user avatar
3 votes
1 answer
106 views

When is compactness of fiber components an open condition?

Consider a smooth map $f:M\rightarrow N$ between smooth manifolds. Ehresmann's theorem states that if $f$ is a proper submersion, then it is locally trivializable; in particular, this implies that ...
Nikhil Sahoo's user avatar
  • 1,107
9 votes
1 answer
370 views

Do compactly generated spaces have a more direct definition?

Is there an elementary way to define Haussdorf-compactly generated weakly Hausdorff topological spaces in a way that does not need defining topological space first? Weakly Hausdorff sequential spaces ...
saolof's user avatar
  • 1,733
1 vote
0 answers
56 views

Poisson equations for tensors on compact Riemannian manifold

Let $({M},g)$ be a compact Riemannian manifold with Levi-Civita connection $\nabla$. It is well known that the Poisson equation $$\Delta f=S$$ where $\Delta:C^{\infty}({M})\to C^{\infty}({M})$ denotes ...
B.Hueber's user avatar
  • 701
4 votes
1 answer
185 views

Is every 1-Lipschitz homeomorphism $f:X\to X$ from a compact metric space to itself an isometry?

I found a statement involving a homeomorphism $f:X\to X$ of a compact metric space $X$, with Lipshitz coefficient 1, i.e., a non-expansive map, and cannot think of an example where $f$ is not an ...
Saúl RM's user avatar
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0 votes
0 answers
70 views

Compactly contained subset with Sobolev functions

I recently asked the very same question on math.stackexchange but unfortunately nobody answered thus far and I would "need" an answer rather quickly, so first of all sorry for doubly posting ...
HelloEveryone's user avatar
0 votes
0 answers
36 views

Perfectly normal compactification of weak-star dual of Banach space

Let $X$ be an infinite-dimensional (otherwise the answer to my question below is trivial) separable real Banach space with topological dual $X^*$, and denote by $\sigma(X^*,X)$ the weak-star topology ...
weirdo's user avatar
  • 101
0 votes
0 answers
132 views

Implicit function theorem on curves

I am trying to figure out, whether the IFT can be generalized to curves. Let's say I have a function $G(x,u)$ mapping $\mathbb{R}^{n+m}\rightarrow \mathbb{R}^n$ with invertible jacobian $\frac{\...
Matthias Himmelmann's user avatar
4 votes
0 answers
124 views

A Lipschitzian's condition for the measure of nonconvexity

I'm actually working on the measure of nonconvexity and its application. Especially, the Eisenfeld–Lakshmikantham MNC defined - in a Banach space - by: $$\alpha(A)=\sup_{b\in\overline{\operatorname{...
Motaka's user avatar
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5 votes
0 answers
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Does "achieving more GH-distances than some compact space" imply compactness?

Previously asked and bountied at MSE: For complete metric spaces $X,Y$, write $X\trianglelefteq Y$ iff for every complete metric space $Z$ such that the Gromov-Hausdorff distance between $X$ and $Z$ ...
Noah Schweber's user avatar
11 votes
2 answers
358 views

When is a k-space locally compact?

We're looking at the possible cardinal sequences of LCS (locally compact, Hausdorff, scattered) spaces, which has led us to think about taking a quotient of a locally compact, scattered space. A k-...
Carla Simons's user avatar
0 votes
0 answers
73 views

A question about a class of pro-$\mathcal X$-group

This question concerns the following lemma of this paper: Lemma 2. Let $\mathcal X_1,\ldots,\mathcal X_n$ be classes of finite groups closed with respect to normal subgroups and subdirect products ...
Meisam Soleimani Malekan's user avatar
4 votes
1 answer
142 views

Integral operator (compactness)

I am studying the compactness of some convolution operators. Let the convolution $$ \Gamma: X\longrightarrow X; x\mapsto\int_0^t T(t-s)B(s)x\mathrm{d}s. $$ Here $T(\cdot)$ is a $C_0$-semigroup on some ...
Malik Amine's user avatar
2 votes
1 answer
110 views

Example of a compact operator that is not uniformly continuous

I want to find a Banach space $E$ and a compact operator $K:[0,1]\times E \rightarrow E$ (that is, $K$ maps every bounded sequence onto a sequence that converges up to a subsequence) satisfying the ...
Vinicius Ramos's user avatar
2 votes
1 answer
374 views

(Dis)prove : if every function with closed graph are continuous then the target space is compact

$(X, \tau_X) $ and $(Y, \tau_Y) $ be two topological spaces. $\forall f\in Y^X$ with $\text{Gr}(f) $ is closed implies $f\in C(X, Y) $. Question : Does this implies $(Y, \tau_Y) $ is compact? ...
Sourav Ghosh's user avatar
4 votes
0 answers
85 views

Reflexivity for the compact open topology on a topological $\mathbb{R}$-vector space

On page 20 of these lectures, Peter Scholze and Dustin Clausen show that a broad class of topological vector spaces is reflexive, i.e. $V \cong [[V, \mathbb{R}], \mathbb{R}]$, where we endow these hom-...
Cayley-Hamilton's user avatar
6 votes
2 answers
236 views

Is every compact simply-connected reductive p-adic group perfect?

Let $k$ be a nonarchimedean local field and $G$ a reductive $k$-group, which we assume to be semisimple and simply-connected. Recall that an abstract group $H$ is perfect if it is generated by ...
David Schwein's user avatar
2 votes
0 answers
140 views

Compactness of a nonlinear operator

Let $H^{1}_{0}(0;\pi)=\{f\in L^{2}(0; \pi): f^{\prime}\in L^{2}(0; \pi)\ \text{and}\ f(0)=f(\pi)=0 \} .$ equipped with the following norm $$\|f\|=\Big(\int_{0}^{\pi}|f'(x)|^2dx \Big)^{\frac{1}{2}}$$ ...
Jaouad's user avatar
  • 31
0 votes
0 answers
97 views

Prove or disprove the compactness of an operator

Consider $X=L^{2}(0,\pi, \mathbb{R})$. Let $X_{\frac{1}{2}}$ be the domain of $(\Delta)^\frac{1}{2}$ where $\Delta$ is the laplacien operator. We define the operator $K:C([0,a],X_{\frac{1}{2}})\...
Jaouad's user avatar
  • 31
0 votes
0 answers
106 views

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 ...
mnmn1993's user avatar
  • 208
9 votes
3 answers
575 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 ...
Corey Bacal Switzer's user avatar
1 vote
0 answers
39 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 ...
wlad's user avatar
  • 4,511
1 vote
0 answers
140 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 ...
Guy Fsone's user avatar
  • 973
0 votes
1 answer
40 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 ...
Federico Vigolo's user avatar
1 vote
1 answer
209 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 $C \subset Z$ ...
kaba's user avatar
  • 387
0 votes
1 answer
120 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 ...
Tomasz Kania's user avatar
5 votes
0 answers
234 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 ...
Cayley-Hamilton's user avatar
4 votes
2 answers
395 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 ...
D.R.'s user avatar
  • 419
7 votes
2 answers
394 views

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 ...
Saúl RM's user avatar
  • 7,164
6 votes
1 answer
702 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. ...
AlexIvanov's user avatar
-1 votes
1 answer
112 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 ...
Motaka's user avatar
  • 291
2 votes
0 answers
115 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 ...
Motaka's user avatar
  • 291
3 votes
0 answers
86 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\...
Meisam Soleimani Malekan's user avatar
0 votes
1 answer
48 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 $Ker \mu=\{X\in \mathfrak{M}_E$ ...
Motaka's user avatar
  • 291
1 vote
0 answers
122 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 ...
Fuutorider's user avatar
4 votes
1 answer
134 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_{...
Meisam Soleimani Malekan's user avatar
1 vote
1 answer
102 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 ...
Rex Lee's user avatar
  • 13
1 vote
1 answer
109 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 ...
user avatar
0 votes
2 answers
315 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 ...
Fuutorider's user avatar
4 votes
1 answer
201 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 ...
saolof's user avatar
  • 1,733
5 votes
1 answer
150 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 $\alpha,\beta\in \mathcal{X}$, $\...
Sunrit's user avatar
  • 59
0 votes
1 answer
180 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 ...
Malik Amine's user avatar
6 votes
2 answers
694 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 ...
Ben's user avatar
  • 898
1 vote
0 answers
338 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 ...
Malik Amine's user avatar
4 votes
0 answers
149 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 ...
Jack L.'s user avatar
  • 1,415
5 votes
1 answer
418 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(\...
Guy Fsone's user avatar
  • 973
1 vote
1 answer
125 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 ...
IJM98's user avatar
  • 261
1 vote
0 answers
134 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 ...
edamondo's user avatar
  • 111
2 votes
1 answer
177 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 ...
Ma Joad's user avatar
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