All Questions
Tagged with compactness fa.functional-analysis
64 questions
5
votes
1
answer
206
views
Compactness in trace class operators space
Let $H$ be a separable Hilbert space. Let $L_1$ denote the space of trace class operators on $H$ with the trace-class norm $\|\cdot\|_1$, i.e. $\|K\|_1=Tr|K|$ for all $K\in L_1$.
Are there easy ...
- 59
1
vote
1
answer
215
views
Compactness with respect to topology induced by total-variation distance
I've been working on a problem and at some point in the proof I need to show that the following set $$\left\{\mu \in \mathcal{P}_{ac}(\mathbb R^d): \int \varphi(x)\mu(\mathrm{d}x)\leq C\right\}$$
is ...
- 21
3
votes
0
answers
147
views
Embeddings of Bochner-Sobolev spaces with second time derivative
NOTE: I also asked this question here in MSE.
In the weak theory of evolution PDEs, the Bochner-Sobolev spaces are frequently used. For $a,b \in \mathbb{R}$ and $X,Y$ banach spaces, we define these ...
- 91
0
votes
0
answers
49
views
Existence of sequence of regular projections
Reading the book :Krasnosel'skii, M.A.; Pustylnik, E.I.; Sobolevskii, P.E.; Zabreiko, P.P. (1976), Integral Operators in Spaces of Summable Functions, Leyden: Noordhoff International Publishing, 520 p....
0
votes
0
answers
72
views
Sequential compactness via Arzela-Ascoli theorem for uniform state spaces
Let $X$ be a uniform topological space and $C([0,1],X)$ the space of continuous functions from [0,1] to $X$. Assume that for subsets of $X$ sequential compactness and compactness are equivalent. Let $(...
- 133
0
votes
0
answers
89
views
Weakly compact set
I want to show that if the set
$$
\big\{u \in L^{q}([0, n] ; X): u(t) \in \phi(t, x(t)), t \in[0, n]\big\}
$$
is weakly compact, then the set
$$
\mathcal{S}_{\phi}(x)=\Big\{u\in L_{loc}^{q}(\mathbb{R}...
- 39
1
vote
0
answers
76
views
Uniform approximation over compacts using weighted function spaces
I'm interested in approximations over the so-called weighted function spaces.
Let $(X,\tau_X)$ be some completely regular Hausdorff topological space. Additionally, consider some map $\psi: X \to (0,\...
- 161
5
votes
0
answers
94
views
When a compact subset of a TVS can be continuously projected on a closed linear subspace?
Let $V$ be a (Hausdorff) topological vector space, $W\subset V$ a closed linear subspace, $X\subset V $ a compact.
(Q):
When there is a continuous map $P:X\to W$ such that $P(x)=x$ for every $x\in X\...
- 60.5k
11
votes
1
answer
309
views
Which closed subsets $Y$ of a compact space $X$ admit a linear extensor $C(Y)\to C(X)$?
In the following $X$ is a Hausdorff compact topological space. Let $Y$ be a closed subset of $X$.
The restriction operator $R_Y:C(X)\to C(Y)$ is surjective (Tietze), so it admits a continuous right ...
- 60.5k
0
votes
0
answers
63
views
Existence of a measurable maximizer
Let $F$ be a continuous cdf with full support on $[0,1].$ Let $A$ be a compact subset of $\mathbb{R}$ and $\mathcal{M}$ be the set of measurable functions $\alpha:[0,1]\rightarrow A.$ Let $\bar \alpha ...
- 1
2
votes
0
answers
170
views
finite dimensionality of a subspace of a Banach space
Let $H$ be the space of measurable functions on $(0,1)$ such that
$$ \|u\|_{H}^2 = \int_0^1 x^2\,|\partial_x u|^2\,dx + \int_{0}^1 |u|^2\,dx <\infty.$$
Let $C>0$ be a constant. Suppose that $W \...
- 4,135
2
votes
1
answer
159
views
A compact embedding claim
Let $U= (0,1)\times (0,1)$. Consider the weighted Sobolev spaces $H_1$ with the norms
$$ \|u\|_{H_1}^2 = \int_0^1 (\int_0^1 x\,|u(x,y)|^2\,dx) \,dy$$
Let $H_2$ be the weighted Sobolev space with the ...
- 4,135
6
votes
3
answers
962
views
Convolution of $L^2$ functions
Let $u\in L^2(\mathbb R^n)$: then $u\ast u$ is a bounded continuous function. Let me assume now that $u\ast u$ is compactly supported. Is there anything relevant that could be said on the support of $...
- 16.2k
4
votes
0
answers
132
views
A Lipschitzian's condition for the measure of nonconvexity
I'm actually working on the measure of nonconvexity and its application. Especially, the Eisenfeld–Lakshmikantham MNC defined - in a Banach space - by:
$$\alpha(A)=\sup_{b\in\overline{\operatorname{...
- 291
2
votes
1
answer
184
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 ...
4
votes
0
answers
109
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-...
user30211
2
votes
0
answers
266
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 $$\|f\|=\Big(\int_{0}^{\pi}|f'(x)|^2dx \Big)^{\frac{1}{2}}$$
...
- 39
0
votes
0
answers
150
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}})\...
- 39
1
vote
0
answers
292
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 ...
- 1,101
5
votes
1
answer
805
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 ...
- 831
-1
votes
1
answer
120
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 ...
- 291
2
votes
0
answers
125
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 ...
- 291
1
vote
1
answer
141
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 ...
- 13
1
vote
1
answer
123
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 ...
user139844
0
votes
1
answer
269
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 ...
- 299
1
vote
0
answers
539
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 ...
- 299
4
votes
0
answers
202
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 ...
- 1,453
5
votes
1
answer
518
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(\...
- 1,101
1
vote
0
answers
137
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 ...
- 111
2
votes
0
answers
1k
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:=\...
- 657
5
votes
1
answer
490
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,$...
- 3,425
8
votes
1
answer
207
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 ...
- 81
1
vote
0
answers
55
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 ...
- 117
1
vote
1
answer
137
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 ...
- 199
5
votes
1
answer
923
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 ...
- 585
0
votes
0
answers
132
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 ...
- 253
0
votes
1
answer
393
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 ...
- 1,955
2
votes
0
answers
2k
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 ...
- 364
2
votes
1
answer
135
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 ...
- 291
-1
votes
1
answer
102
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(...
- 617
7
votes
1
answer
856
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 ...
- 395
0
votes
0
answers
172
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}...
- 31
2
votes
1
answer
246
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 ...
- 163
4
votes
2
answers
244
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 ...
- 41.8k
7
votes
1
answer
850
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 ...
- 105
1
vote
0
answers
196
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 $L^2(0,\tau;H^2(0,\ell)\cap H^1_0(0,\ell))$...
- 395
0
votes
2
answers
1k
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, ...
- 395
5
votes
1
answer
506
views
Weak compactness of order intervals in $L^1$
Let $(\Omega,\mu)$ be a measure space, say $\sigma$-finite for the sake of simplicity, and let $L^1 := L^1(\Omega,\mu)$ denote the real-valued $L^1$-space over $(\Omega,\mu)$.
For all $f,h \in L^1$ ...
- 12.5k
10
votes
0
answers
201
views
Masas in SAW*-algebras
I asked this question three years ago at MSe but it has no response; let me try here.
Pedersen distilled the following class of C*-algebras which he termed SAW*-algebras (Journal of Operator Theory, ...
- 11.3k
5
votes
2
answers
852
views
Covering compactness in the weak sequential topology
Let $X$ be a real Banach space. Apart from the norm topology, we can consider the following weak topologies on $X$:
the weak toplogy, defined as the initial topology with respect to $X^*$. In other ...
- 171