All Questions
1,591 questions
1
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81
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Relation between minimizer of regularized risk & risk in statistical learning theory
In supervised machine learning, we typically take a Risk Minimization (RM) point of view when formulating a problem. So, what we typically solve for is the following:
$$ R^L(h) = \underset{h\in\...
- 11
1
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0
answers
110
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Properties of Sobolev spaces $W^{k,p}(\Omega, E)$ where $E$ is a Banach space
$\newcommand{\R}{\mathbb R}$Let $E$ be a Banach space with norm $\|\cdot\|_E$ and let $\Omega\subset \R^n$ be an open set.
For $k\geq 0, p\geq 1$ we define $W^{k,p}(\Omega, E)$, the Sobolev space of ...
- 2,533
1
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1
answer
452
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Uniformly convex Banach spaces
Theorem. If $X$ and $Y$ are uniformly convex Banach spaces, then for $1<p<\infty$ the space
$$
X\oplus_pY=X\times Y,
\qquad
\Vert(x,y)\Vert:=(\Vert x\Vert_X^p+\Vert y\Vert_Y^p)^{1/p}
$$
is ...
- 28k
1
vote
2
answers
273
views
A Borel perfectly everywhere surjective function on the Cantor set
Does there exist a Borel (or even continuous) function $f:\mathcal{C}\to\mathcal{C}$, where $\mathcal{C}$ is the Cantor set (or Cantor space $2^\omega$) such that for every nonempty closed perfect set ...
- 3,115
1
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1
answer
114
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Is a $1_A \otimes U$ invariant subspace of $\mathcal{H}_A \otimes \mathcal{H}_B$ a product $V_A \otimes \mathcal{H}_B$?
Consider a subspace $V$ of $\mathcal{H}_A \otimes \mathcal{H}_B$, with $\mathcal{H}_A$ and $\mathcal{H}_B$ finite-dimensional Hilbert spaces, that is $1_A \otimes U$ invariant for all unitary ...
- 133
1
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1
answer
187
views
Fractional Laplacian equation on a ball and explicit solutions
Let us consider
\begin{align*}
(-\Delta)^s u &= 0 && x \in B_r(0) \\
u&=0 && x \in \mathbb R^N \setminus B_r(0),
\end{align*}
where $$
(-\Delta)^s u(x) = \int_{\mathbb{R}^N} \...
- 99
1
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0
answers
346
views
Duality of maps on bounded vs trace-class operators (Schrödinger-Heisenberg dual)
$\newcommand\calH{\mathcal H}
\newcommand\calK{\mathcal K}
\newcommand\tr{\operatorname{Tr}}$I am looking for a (citable) reference for the following fact:
Bounded linear maps $g:T(\calH)\to T(\calK)$...
- 896
1
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0
answers
150
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Lifting theorem for finite spaces: replacing perfect normality by normality
In the Lifting theorem for finite spaces (Thm. 3.5, Eric Wofsey, quoted below),
can one relax the condition "$A$ is a closed subset of a perfectly normal $X$" to
"$A\to X$ has the right ...
- 317
1
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1
answer
129
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extending disjoint open subsets of a normal Hausdorff space
Under what assumptions on $C$ and $X$ is the following true ? I was neither able to find a counterxample or prove this, though it appears that compactness, e.g. assuming $X$ is compactly generated, ...
- 81
1
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1
answer
426
views
$L^p$ compactness for a sequence of functions from compactness of cut-off
Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
- 161
1
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1
answer
511
views
Convergence in $C_c$ but not in $C$
Let $C_c(\mathbb{R})$ be the set of compactly-supported continuous functions on $\mathbb{R}$. We can view this with a number of different topologies but I have my eye on two in particular. Let $X$ ...
- 5,405
1
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0
answers
177
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Building random homeomorphisms of the torus $\mathbb T^2$
In https://arxiv.org/abs/0912.3423, a family of random homeomorphisms of the circle is constructed. Main Question: Can the construction be generalized to higher space dimensions, e.g. to $\mathbb T^2$?...
- 101
1
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3
answers
435
views
Operator norm of shift operator for finite measure spaces
Let $\nu$ be a finite Borel measure on $\mathbb{R}^n$ and define the shift operator $T_a$ on $L^p_{\nu}(\mathbb{R}^n)$ by $f\to f(x+a)$ for some fixed $a\in \mathbb{R}^n-\{0\}$. Suppose moreover that ...
- 5,405
1
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0
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53
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Spaces that are comparable with their compacts
This is an outgrowth of this question.
For a (metrizable) space $X$ consider the following increasingly strong properties:
(i) For every compact $K\subset X$ there is a map $f:X\to X$ such that $K\...
- 5,529
1
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1
answer
387
views
$L^p$ compactness for a sequence of functions from compactness of product with cut-off
Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
- 161
1
vote
1
answer
189
views
Complemented subspaces in a dual Banach space
Let $Y$ be a complemented subspace in a dual Banach space $X$. Is it true that $Y$ is itself isomorphic to a dual?
This is the case of a $w^*$-closed subspace $Y$, but a complemented subspace of $X^*...
- 60.5k
1
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0
answers
824
views
A weakly sequentially continuous operator which is not weakly continuous
I'm reading some papers where the condition of weak sequential continuity is crucial instead of the weak continuity.
So, let
$T$ an operator between a Banach space $X$ and itself.
$T$ is weakly ...
- 291
1
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0
answers
178
views
A locally convex $C^*$ algebra without zero divisor
Let we have a locally convex $C^*$ algebra $A$. That is $A$ is a TVS equipped with an algebra and an involution structure such that all operations are continuous. Moreover the topology on $A$ ...
- 356
1
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0
answers
393
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Unambiguous "weak" vector valued $L^{+\infty}$ spaces?
For some time, I have been stuck to the problem to be described as follows. The (perhaps not so commonly known) facts given here are taken from R. E. Edwards' Functional Analysis (Holt, Rinehart and ...
- 3,584
1
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1
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82
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What is $\left\| u \right\|_{ H_{0}^{k}, H_{0}^{k}}$ norm when $H_{0}^{k}=\left\{u \in H^{k, 2}(M) \mid \int_{M} u \operatorname{vol}_{g}=0\right\}$
What is $\left\| f \right\|_{ H_{0}^{k}, H_{0}^{k}}$ norm when $H_{0}^{k}=\left\{u \in H^{k, 2}(M) \mid \int_{M} u \operatorname{vol}_{g}=0\right\}$.
I'm reading a paper
Chern-Yamabe flow
which said ...
- 809
0
votes
1
answer
216
views
Complex Borel measures: relation between the total variation norm of a measure and those of its real and imaginary parts
Let $X$ be a metric space and $\mathcal B$ its Borel $\sigma$-algebra. For $B \in \mathcal B$ we denote by $\Pi(B)$ the collection of all finite measurable partitions of $B$, i.e.,
$$
\Pi(B)=\left\{\...
- 657
0
votes
1
answer
365
views
Integral in a σ−convex set.
Having had no (proper) answer to this question, I formulate the remaining case as a new question as follows. With $I=[0,1]$, let $E$ be a separable (real) Banach space, and let $\gamma:I\to E$ be ...
- 3,584
0
votes
1
answer
77
views
Decay rate of minimum point over a product space
Let $f(\theta, \epsilon)$ be smooth on $[0,2\pi] \times [0,\infty)$ such that
$f(\theta, \epsilon)$ converges to $f(\theta, 0)$ uniformly as $\epsilon \rightarrow 0$.
$f(\theta, \epsilon) > 0$ for ...
- 434
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1
answer
491
views
Is this set of function belongs to $L^\infty$?
Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $u\in SBV\cap L^\infty(\Omega)$ be given. We write
$$
Du = \nabla u\lfloor \mathcal L^N + (u^+-u^-)\otimes \nu_u\mathcal H^{N-...
- 679
0
votes
1
answer
127
views
Continuous extensions of tangent vector fields
Let $\Omega$ be an open subset of $S^2$ with $\bar{\Omega}\neq S^2$. Suppose a continuous tangent vector field $G$ is given on $\partial \Omega$ with $|G(y)|=1$ for all $y\in \partial \Omega$. Does ...
- 434
0
votes
1
answer
53
views
Rate of convergence of the minimum point over a product space
Let $f(\theta, \epsilon)$ be smooth on $[0,2\pi] \times [0,\infty)$ such that
$f(\theta, \epsilon)$ converges to $f(\theta, 0)$ uniformly as $\epsilon \rightarrow 0$.
$f(\theta, \epsilon) > 0$ for ...
- 434
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votes
1
answer
88
views
Is the mapping $F(a):= \arg\min_{x \in \mathbb R^n} \|x-a\|_2 + \|x\|_1$ non-expansive?
Fix $a \in \mathbb R^n$ and let $\|\cdot\|$ be any norm on $\mathbb R$ (e.g $\ell_1$ norm). For any $a \in \mathbb R^n$, it is clear that the function $f_a(x) := \|x-a\|_2 + \|x\|$ is strictly convex ...
- 6,853
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0
answers
173
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Is this has anything to do with Riesz representation?
The Riesz representation is very useful in study BV space. There is a lot of version of it and one of the good one can be found in this book, page 49.
Here I come up with a question which has similar ...
- 679
0
votes
1
answer
78
views
The conditions used to prove upper semicontinuous of generalized directional derivative (in Clarke sense)
Let $X$ be a reflexive Banach space, $z, x, v \in X$. $\{z_i\}, \{x_i\}$ and $\{v_i\}$ are arbitrary sequences converging to $z, x$ and $v$, respectively.
I would like to know under which conditions ...
- 15
0
votes
0
answers
127
views
Approximation Property: Characterization
Problem
Given a Banach space $E$. Denote compact sets by $\mathcal{C}$, compact operators by $\mathcal{C}(X,Y)$, and finite rank operators by $\mathcal{F}(X,Y)$.
Suppose it has the approximation ...
- 598
0
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0
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302
views
Banach space of discontinuous functions on a product space
Edit: According to comments of Eric Wofsy and Yemon Choi I edit the question.
For a (compact) topological space $X$, we put $A=\{f:X\to \mathbb{C}\mid f\text{ is bounded}\}$. We define a semi-...
- 356
0
votes
1
answer
125
views
Bounding integral expression with Sobolev norm of integrand
Consider the following integral expression:
$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$
for $\epsilon>0$, $f \in L^\...
user139844
0
votes
1
answer
172
views
Can we build a continuous function from "fibers"/preimages defined over a topological base?
I am looking for some proof insight or literature references for a statement which, if it's actually true, is probably a pretty trivial thing. I hope the question has not been asked here before, and ...
- 273
0
votes
0
answers
171
views
Explanation of a step in a preprinted work
I have been studying this preprinted paper and the references therein. I believe that there are some typos; However, since I am not an expert yet, I would like to make sure I am correct.
I do not ...
- 159
0
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0
answers
114
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Recontruction of the weak topolgy from the scalar product on a subset of a Hilbert Space
Let $M$ be a set a let $K:M\times M\to\mathbb{C}$ be a positive definite kernel. By a version of Moore-Aronszajn Theorem, there is a unique (up to the unitary euivalence) Hilbert Space $X$, and a map $...
- 5,529
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votes
1
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697
views
How much do we know about this "local" Hardy-Littlewood maximal function?
The "local" Hardy-Littlewood maximal function is given by $$(M_\phi f)(x)= \sup_{0<\epsilon<1}|\phi_\epsilon \ast f|(x),$$ which is similar to the classical Hardy-Littlewood maximal function : $$...
- 171
0
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0
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113
views
Finite sets are residual in the Hausdorff space
Let $X$ be a metric space, let $\mathbb{H}(X)$ denote the set of non-empty closed subsets of $X$ with Hausdorff metric which we denote by $d_{\mathbb{H}(X)}$, and let $\mathbb{H}_{\operatorname{fin}}(...
- 5,405
0
votes
2
answers
230
views
Basic sequences in $ L_{p}$
Let $(x_{n})_{n}$ be a normalized basic sequence in $X=L_{p}$, with $1<p<2$.
Does there exist a subsequence $(x_{k_{n}})_{n}$ of $(x_{n})_{n}$ and a weakly null sequence $(x^{*}_{n})_{n}$ in $X^...
- 3,343
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votes
1
answer
74
views
$T_2$-spaces such that the lattices of open sets can be embedded into each other
Let $(X,\tau), (Y,\sigma)$ be $T_2$-spaces such that there are injective lattice homomorphisms $f: \tau\to \sigma$ and $g:\sigma\to \tau$.
Does this imply that $(X,\tau)\cong (Y,\sigma)$?
user62017
0
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0
answers
118
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A measure on the group of homeomorphisms of $\mathbb T^2$
Let us consider the group of measure-preserving homeomorphisms of $\mathbb T^2$ (with transformations identified if they agree almost
everywhere) called $G[\mathbb T^2, \mathcal L^2]$. We shall ...
- 101
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votes
1
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98
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Is every subgroup closed in this complete, nondiscrete topological group?
Another question on Mathoverflow (here: Complete topological groups in which all subgroups are closed) asks if there exists a complete, nondiscrete topological group $G$ such that all subgroups of $G$...
- 361
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votes
1
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236
views
Is this a contraction mapping for small $T$?
Let $G$ be the heat kernal, i.e. for $0\le t<s$ and $x,y\in\mathbb R$
$$G(t,x;s,y):=\frac{1}{\sqrt{4\pi(s-t)}}\exp\left(-\frac{(y-x)^2}{4(s-t)}\right).$$
For $T>0$, let $\mathcal H_T:=\{h:[0,T]\...
- 1,334
0
votes
0
answers
112
views
Fixed point of a contraction map
This question is a continuation of Is this a contraction mapping for small $T$?
Set, for $T, m>0$, $H^m_T:=\{h:[0,T]\to [0,m]:~ h,~h' \mbox{ are both continuous on } [0,T]\}$ endowed with the norm
$...
- 1,334
0
votes
1
answer
238
views
Minimal totally separated spaces
Let us call a space $(X,\tau)$ totally separated (ts) if for every two distinct points there is a clopen set containing one, but not the other. If for every topology $\sigma\subseteq\tau$ with $\sigma\...
- 48.9k
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votes
1
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233
views
Does every compact countable space contain a non-trivial convergent sequence?
Problem. Does every compact countable space contain a non-trivial convergent sequence?
This question concerns non-Hausdorff compact spaces. An example of such space is any infinite set $X$ endowed ...
- 41.9k
0
votes
2
answers
287
views
Distinguishable under manifold topology but indistinguishable under the Alexandrov topology
Take the time-oriented Lorentzian spacetime $(M, g)$ that is not strongly causal.
In such case it is shown that the Alexandrov topology and the Manifolds topology deviate such that the manifold ...
- 612
0
votes
1
answer
116
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Integrable function [closed]
Suppose that $a, b, c_1$ and $c_2$ are real constant.
Is there the necessary and sufficient conditions of $a ,b, c_1,c_2 $ for the following integration is integrable? i.e.
$$\int_1^{\infty}\int_1^{\...
- 119
0
votes
1
answer
155
views
Does the set of positive definite kernels on some set X form a ring?
Given some non-empty set $X$, does the set of positive definite kernels on $K_X$ form a ring with pointwise addition and multiplication. I am convinced it does not as surely if $k \in K_X$ then we ...
0
votes
1
answer
110
views
Functions for which $|f^{(k)}|_{C^{0,\alpha}(0,1)} \le \Vert f \Vert_{L^1(0,1)}$
Let $f \in C^k(0,1)$ and assume that the $k$-th derivative is $\alpha$-Hölder continuous. Assume that $f(x) = 0$ in a fixed interval $(a,b) \subset (0,1)$. Can we characterize (or at least find some ...
- 131
0
votes
2
answers
545
views
Suppose $(G,\mathcal T)$ is a paratopological group and $a,b\in G$ and every neighborhood of $a$ contains $b$. Can we say every neighborhood of $b$ contains $a$?
Suppose $(G,\mathcal T)$ is a paratopological group and $a,b\in G$ and every neighborhood of $a$ contains $b$. Can we say every neighborhood of $b$ contains $a$?
clearly every closed neighborhood ...
user31967