All Questions
10,448 questions
0
votes
1
answer
117
views
Existence of sequence of distributions
This question concerns distributions $\mu$ over the naturals $\mathbb{N}=\{1,2,\ldots\}$. For $q\ge1$, let us define the $q$th moment of entropy:
$$
H_q(\mu)=\sum_{i=1}^\infty \mu(i)|\log\mu(i)|^q,
$$
...
- 6,541
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
323
views
Is the restriction map $C^1\ni f\mapsto\left.f\right|_K$ a continuous map?
Let $E$ be a $\mathbb R$-Banach space, $\Theta\subseteq C^{0,\:1}(E,E)$ be a $\mathbb R$-Banach space and $\iota$ be a continuous embedding of $\Theta$ into $C^1(E,E)$.
I would like to show that, ...
- 167
0
votes
1
answer
289
views
Ellipsoid in $L^p([0,1],\lambda)$ spaces?
Let us consider $L^p([0,1],\lambda)$ spaces, were $\lambda$ is simply the lebesgue measure. These are Banach spaces for $p\ge1$ (of course). It is well known that for $ 1\leq p < q \leq +\infty$ we ...
user153000
0
votes
1
answer
150
views
Expectation of random matrix
Assume $Q$ is a positive definite random matrix such that $0 < \lambda_{\min}(Q)....\leq \lambda_{\max}(Q) \leq 1$ holds. I want to show that
\begin{align}
E\left[\frac{\lambda_{\min}(Q)}{\lambda_{\...
- 3
0
votes
1
answer
322
views
Injectivity of analytic functions
Suppose $f : \mathbb{R} \rightarrow \mathbb{R}^n$ is a real analytic function on $(a, \infty)$. I have two questions:
Suppose $||f(x)|| \rightarrow \infty$ as $x \rightarrow \infty$. I know without ...
- 431
0
votes
1
answer
81
views
If $\tau_1\subset \tau_2$ and $X^*$ is separable for $\tau_1$ then $X^*$ is separable for $\tau_2$?
Let $X$ be a Banach space the associated dual space is denoted by $X^*$. Take $\tau_1$ and $\tau_2$ two topologies in $X^*$ compatible with the duality $(X^*,X)$, such that $\tau_1\subset \tau_2$.
...
- 199
0
votes
1
answer
102
views
Is the integral functional $I(x) = \int_{0}^{T} \Lambda (t , x(t), \dot{x} (t)) \; dt $ locally lipschitz on the space $C^2 [0 ,T] $?
Let the function $\Lambda : [0,T] \times \mathbb{R^n} \times \mathbb{R^n} \to \mathbb R$ be continuously differentiable. Then the integral functional $I(x) = \int_{0}^{T} \Lambda (t , x(t), \dot{x} (t)...
- 369
0
votes
1
answer
149
views
Cores in the tensor-train decomposition
Let $d_i\in\mathbb N$, $I_i:=\{1,\ldots,d_i\}$ and $u\in\mathbb R^{d_1}\otimes\mathbb R^{d_2}\otimes\mathbb R^{d_3}$. It's somehow clear to me that we may regard $u$ as a three-dimensional array (see ...
- 167
0
votes
1
answer
114
views
$ \overline{(A-A)}\cap\overline{B}(0,r)\text{ is weakly compact, }\forall r>0 $?
Let $X$ be a separable Banach space and $A$ is a subset of $X$ such that
$$
A\cap\overline{B}(0,r) \text{ is weakly compact, } \forall r>0.
$$
Can we say that :
$$
\overline{(A-A)}\cap\overline{...
- 117
0
votes
1
answer
1k
views
Bounding $L^p$ norms in terms of lower-order $L^q$ norms
Suppose $f,g\in L^q(\Omega)$ ($\Omega\subset \mathbb{R}^n$) for all $1\le q\le p$. Here, $L^p(\Omega)$ is defined with respect to some measure $\mu$ that is absolutely continuous wrt Lebesgue measure. ...
- 710
0
votes
1
answer
158
views
Showing a product on a character space is continuous
Quoting from Timmermann's An invitation to quantum groups and duality:
Prop. 5.1.3 Let $A$ be a commutative algebra of functions on a compact
quantum group. Then there exists a compact group $G$ and ...
- 1,037
0
votes
1
answer
91
views
Choosing the best submatrix
Let $\mathbf{A}_{m\times n}$ be a matrix with non-negative elements. Assume that a submatrix $\mathbf{B}$ from $\mathbf{A}$ is defined as
\begin{align}
B_{i,j} =
\begin{cases}
A_{i,j}, & i\in\...
- 287
0
votes
1
answer
752
views
Real part of entire function property
Is there any characterization of the set of entire functions $f(z) $ such that $\Re(f(z)) \geq \Re(\overline{f(\bar{z})})$ for all $z\in \mathbb{C}^{+} $?
($\Re$ stands for the real part)
Edit: I ...
- 39
0
votes
1
answer
407
views
Criteria for $\epsilon$-Density
Let $Y$ be a compact, separable metric space and $X=C(Y)$ Banach space. There are many criteria for a linear subspace $Z\subseteq X$ to be dense; notably the Stone-Weierstraß theorem.
Are there ...
- 5,405
0
votes
1
answer
359
views
Dual norm of a max function [closed]
I am attempting to find the dual norm of
$$\|(x,y)\|_K=\max\{|x|,|y|,|x-y|\}.$$
I have obtained $\|(x',y')\|_K^* = |x'|+|y'|$, but don't think that this is correct.
I obtained this as follows :
$$K = ...
- 9
0
votes
1
answer
133
views
Product of sets with the Radon-Nikodym Property (RNP)
I have read that it is somewhat well-known that if two Banach spaces $X$ and $Y$ have the Radon-Nikodym Property (RNP), then their product $X\times Y$ also has the RNP.
Does the above result ...
- 1,245
0
votes
1
answer
174
views
Does $\{\left|\varphi\right>\left<\psi\right|+\left|\psi\right>\left<\varphi\right||\varphi\in\{\psi\}^{\perp}\}$ split $\mathfrak{S}_1$?
Let $\mathfrak{S}_1$ be the space of trace-class self-adjoint operators on $L^2(\mathbb{R}^n)$, and $\psi\in L^2(\mathbb{R}^n)$ such that $\int |\psi|^2 = 1$. Is there a projection from $\mathfrak{S}...
user141436
0
votes
2
answers
299
views
Solution of ODE with discontinuity
Let $F:\mathbb{R} \to \mathbb R$ be a bounded Lipschitz function and $G(x,y) = (0,\chi_{\{x \le F(y)\}})$.
Consider the ODE
$$
\begin{cases}
\partial_t \Phi(t,x) = G(\Phi), & t \in [0,T]\\
\Phi(...
- 839
0
votes
1
answer
75
views
Fractal set $E$ such that the indicator function $\mathbf{1}_E$ is BV
Is there a "fractal" set $E \subset \mathbb R^2$ such that the indicator function $\mathbf{1}_E$ is in $BV(\mathbb R^2)$?
- 839
0
votes
1
answer
133
views
Reference request on Borsuk conjecture [closed]
I just heard of Borsuk conjecture. I want to ask if there are any references preferably looking at the problem from the point of view of Mathematical analysis I can study it from?
Thanks
- 139
0
votes
1
answer
220
views
Regarding extreme point in a Banach space
Let $X$ be a Banach space. And let $X^* $ be the dual space of $X$. Let $E_X$ and $E_{X^*}$ denote the extreme points of the unit ball of $X$ and $X^*$. Let $x\in X$ and $|f(x)|=1$ for every $f\in E_{...
- 195
0
votes
1
answer
244
views
Proving that $\|\mathbf{T}^n\|^2=\sum_{g\in \mathbf{G}(n,d)}\|\mathbf{T}_g\|^2\,$
Let $F$ be a complex Hilbert space and $\mathcal{B}(F)$ be the algebra of all bounded linear operators on $F$.
For ${\bf A} = (A_1,...,A_d) \in \mathcal{B}(F)^d$, the norm of ${\bf A}$ is given by
$...
- 724
0
votes
1
answer
419
views
Stone–von Neumann theorem?
The Stone–von Neumann theorem says that given two unitary groups on a Hilbert space $H$ satisfying the canonical commutation relations (CCR)
$$
U(t)V(s) = e^{-i st} V(s) U(t) \qquad \forall s, t
$$
...
0
votes
1
answer
350
views
Uniformly Bounded (updating)
Suppose that $a_1<1$, $a_1+a_2+a_3>1.$ For $x,y,z>0,$
(1) define a fucntion
$$H(x,y,z)=\frac{x^{\frac{1}{2}}\int_0^{\infty}\frac{1}{t^{a_1}~ (1+t)^{a_2+1}~
(1+t+z)^{a_3}}\exp\big\{-\frac{...
- 119
0
votes
1
answer
284
views
Weak convergences in Bochner spaces
I'm having bit trouble in understanding weak convergences in Bochner space. I have following question for some general measurable space $\Omega$:
Let $\{x_n\}$ be a bounded sequence in $L^2((0,T)\...
- 101
0
votes
1
answer
242
views
Harnack inequality for fractional laplacian
Let u be a positive solution of $s\in (0, 1) $
\begin{equation}
\left\{\begin{aligned}
(-\Delta )^{s} u &= 0 \text{ in } (-2T, 2T)\\
u &=g\quad\text{in}\quad \mathbb R\setminus(-2T, 2T).
\...
- 402
0
votes
2
answers
235
views
An inequality on length of two curves [closed]
I am looking for a proof, reference, comment of an inequality as follows:
If $f(x)$ and $g(x)$ be two continuous derivative funcions in interval $[a, b]$. Such that:
$f(a)=g(a)$ and $f(b)=g(b)$
$(...
- 1,776
0
votes
2
answers
170
views
dense subalgebra in measurable functions set
Take $\mathcal M_D$ the space of measurable functions from a compact set $D\subseteq \mathbb R_n$ to ℂ.
I'm wondering if a Stone-Weierstrass-like theorem holds in this space, with the convergence in ...
- 201
0
votes
1
answer
98
views
Fix a continuous function $f:X\times X^k\to Y$ multilinear in $X^k$, for $X,Y$ Banach. Is $f:X\to\mathscr{L}(X,\ldots,X;Y)$ continuous?
Fix two infinite-dimensional Banach spaces $X,Y$. We define the space
$$
\mathscr{L}(X,\ldots,X;Y)=\mathscr{L}(\underbrace{X,\ldots,X}_{k};Y)
$$
to be the set of continuous multilinear operators $T:X^...
- 1,247
0
votes
1
answer
217
views
Finding closed form expression for the roots of $f(x) = \sum_{i=1}^K \frac{\alpha_i \gamma_i \sin(x-\theta_i)}{1+\gamma_i[1+\cos(x-\theta_i) ]}$
Let us define function $f:[0~ 2\pi] \rightarrow R$ as follows:
\begin{align}
f(x)\triangleq \sum_{i=1}^K \frac{\alpha_i \gamma_i \sin(x-\theta_i)}{1+\gamma_i[1+\cos(x-\theta_i) ]},
\end{align}
...
- 105
0
votes
1
answer
269
views
Limit of eigenvalues of a matrix perturbation sequence
Suppose $H$ is an $n\times n$ symmetric positive definite matrix, $M_k$ is a sequence of $n \times n$ matrix (not necessarily symmetric) such that $M_k \to O$ where $O$ is the zero matrix. Let $\...
- 135
0
votes
1
answer
203
views
For $B=\int \lambda d E_\lambda $ and $X$ commutes with every $E_\lambda $, why $BX$ is positive and self-adjoint?
Let $B$ be an unbounded closed operator on a Hilbert space $H$. If $B=\int \lambda d E_\lambda $ is positive self-adjoint and a positive bounded operator $X$ commutes with every $E_\lambda $, then why ...
- 617
0
votes
1
answer
221
views
A weakly open subset of the unit ball of the Read's space $R$ (an infinite-dimensional Banach space) is unbounded
We know every weakly open subset of an infinite-dimensional Banach vector space X is unbounded.
Now, Read's space $R$ (an infinite-dimensional Banach space) has the property:
there is $ρ >0$ such ...
- 93
0
votes
1
answer
136
views
When are Weighted $\mathcal{L}^p$-Spaces Topologically Isomorphic?
Let $X$ be a topological space and $\mu$ be the Borel measure on $X$. Suppose $W_1$ and $W_2$ are continuous, non-negative functions from $X$ into the real numbers such that, for all integers $p > ...
- 263
0
votes
1
answer
237
views
continuity of b-metric
A b-metric is defined similar to a metric in which the triangle inequality is replaced by the inequality
$$d(x,z)\leq s\Big[d(x,y)+d(x,z)\Big]\quad\forall\ x,y,z$$
where $s\geq1$.
There is an example ...
- 149
0
votes
2
answers
653
views
Poincaré inequality on annular regions
It is well known that given a function $f \in L^p(B_R)$ such that $|\{x \in B_R: f(x) = 0\}|>0$, the following Poincaré inequality holds:
$$ \int_{B_R} \left(\frac{|f|}{R}\right)^p \ dx \leq c \...
- 455
0
votes
1
answer
186
views
Meromorphic solutions to Legendre's equation
I just saw the following question that was asked yesterday on math overflow on meromorphic solutions to ODEs
Although, I understand the answers and comments to the questions, I did not understand how ...
- 501
0
votes
2
answers
160
views
A matrix between vectors, and inequality!
I have an inequality as follows
$$s^T\phi\leq -|s|^TA$$
where $s$, $\phi$ and $A$ are vectors with appropriate dimensions. I want to prove that this inequality holds for the following too
$$s^TM\...
- 29
0
votes
1
answer
104
views
Operator identity for convergent series
Let $T_i$ and $S_i$ be a sequence of bounded operators such that
$$ \sum_{k,i,j=0}^{\infty} S_j^* T_i^* T_i S_k$$ converges unconditionally in operator norm on some Hilbert space. The limit is then ...
0
votes
1
answer
103
views
Suppose that $a \mu = \mu a$ for all $a$ in $C^*$-algebra $A$. Then $\mu \in Z(A^{**})$
Let $A$ is a $C^*$-algebra and $\mu \in A^{**}$. Suppose that $a \mu = \mu a$ for all $a \in A$. Then $\mu \in Z(A^{**})$.
- 19
0
votes
1
answer
111
views
If $A$ is a $C^*$-algebra, then $H^1 (A, D(A)) = \{ 0 \}$ (first cohomology group )?
If $A$ is a $C^*$-algebra, then $H^1 (A, D(A)) = \{ 0 \}$ (first cohomology group )?
We don't know that is an open problem or it has counterexample...
- 19
0
votes
1
answer
350
views
The order of companion matrix over various modulo
We consider a positive integer number and call it our modulo and denote it with $m$. We choose a positive integer number like $p$ and
call it the degree of our polynomial. We select $p$ integer ...
- 313
0
votes
1
answer
152
views
Transitivity of the Cuntz sub-equivalence
Let $A$ be a $C^*$-algebra and $a,b \in A$ positive elements. We define a relation (Cuntz sub-equivalence) by saying
$$a\lesssim b: \Leftrightarrow \exists\, (r_n)_{n\in\mathbb{N}}\subset{A}\text{ ...
- 1
0
votes
1
answer
222
views
Differential Riccati-type equation
Setup
I have recently come across an ODE of the form
$$
0 = \dot{A}(t)^TG(t) + A(t)^T\dot{G}(t) + C(t) + \lambda B(t)^TA(t)G(t) + \frac{\lambda}{2} (D(t)A(t)G(t))(D(t)A(t)G(t))^T\bar{1},
$$
where $\...
- 5,405
0
votes
1
answer
317
views
Some questions related to the unitary operators
A unitary operator is a surjective linear operator between complex inner product spaces, which preserves the inner product.
What is the name of the analogue for the real case? Orthogonal operator ...
- 5,529
0
votes
1
answer
843
views
$C^{\infty}_{loc}$-convergence - right definition
Let $\Omega \subset \mathbb{R}^{n}$ be some open set. Let $f_{n},f\in C^{\infty}(\Omega)$. My question is: What does the following phrase mean? $f_{n}$ converges to $f$ in $C^{\infty}_{loc}(\Omega)$. ...
- 35
0
votes
1
answer
128
views
On a theorem by Mooney and Khavin on the weak sequential completeness of the predual of $H^\infty(\mathbb{D})$
There is a theorem by Mooney http://msp.org/pjm/1972/43-2/pjm-v43-n2-p.pdf#page=185 and independently proved by Havin which says that the predual of $H^{\infty}(\mathbb{D}),$
$L^{1}/H^{1}_{0}$ is ...
- 175
0
votes
1
answer
126
views
Nonstable $K$-theory question
Let $Y$ be a compact, Hausdorff topological space, and $X$ be a locally compact, contractible, Hausdorff space which is homeomorphic to a dense subset of $Y$.
Question A: Is $GL_1(C(Y))\stackrel{\pi}...
0
votes
1
answer
214
views
Sobolev chain rule on non-compact manifolds
Let $(M,g)$ be a non-compact Riemannian manifold (not of bounded geometry).
Let $f:\mathbb{R} \to \mathbb{R}$ be $C^1$ with $f'$ bounded and $f(0)=0$. Is the Sobolev chain rule valid for functions $...
- 1