All Questions
13,927 questions
0
votes
1
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138
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Antilinear unbounded operator has closed graph
Let $H$ and $K$ be Hilbert spaces and $D(T)$ a vector subspace of $H$. Let $T: D(T) \to K$ be a densely defined antilinear operator. Its adjoint $T^*: D(T^*)\to K$ is defined by the relation
$$\langle ...
- 175
0
votes
1
answer
233
views
When does $C_b(X)$ admit a Schauder Basis?
Let $(X,d)$ be a separable and connected metric space. My question is rather short and to the point: do there exist $\{x_n\}_{n=0}^{\infty}\subseteq X$ such that
$$
\left\{d(x_n,\cdot)-d(x_0,\cdot)\...
- 195
0
votes
1
answer
151
views
Construction of a probability measure from a sequence of probability measures
Summary
I would like to pass from a sequence of probability measures whose "limit" satisfies a desired property to a new probability measure that satisfies this property.
Details
We work on ...
user475819
0
votes
1
answer
148
views
Proof of vanishing viscosity error rate
Consider the initial value problem associated to the parabolic equation $u^\epsilon_t + u^\epsilon_x = \epsilon u^\epsilon_{xx}$ and the corresponding hyperbolic problem $u_t + u_x = 0$.
What is a ...
user139844
0
votes
1
answer
258
views
Exponential derivative operator and continuous functions
I would like to know how to write down the following expression
$$f(y)=\frac{1}{y^{n} e^{\frac{d}{dy}}g(y)}$$
in the form of $e^{-\frac{d}{dy}}y^{-n}(\frac{1}{g(y)})$ where $n$ is an integer and $f,g: ...
- 159
0
votes
1
answer
86
views
If $\lambda_i$ is symmetric with $\lambda_i\{0\}=0$, why does $\int_B1-\cos\langle x,x'\rangle\:(λ_1-λ_2)({\rm d}x)=0$ imply $λ_1=λ_2$?
Let $E$ be a separable $\mathbb R$-Banach space and $\lambda_i$ be a finite symmetric measure on $\mathcal B(E)$ with $\lambda_i(\{0\})=0$ and $$\int_B1-\cos\langle x,x'\rangle\:\underbrace{(\lambda_1-...
- 167
0
votes
1
answer
513
views
When is a product of two two-parameter Mittag-Leffler functions a Mittag-Leffler function?
I am studying properties of the two-parameter Mittag-Leffler function.
$$ E_{\alpha,\beta}(z)=\sum_{k=0}^\infty \dfrac{z^k}{\Gamma(\alpha k+\beta)}.$$
I am particularly interested in recurrences and ...
- 123
0
votes
1
answer
479
views
Probabilistic interpretation of derivative of a Dirac delta function
Consider $g : \mathbb{R}^d \mapsto \mathbb{R}$ defines some surface $\Sigma$ in $\mathbb{R}^d$. Then I can define a random variable $X_1$ with support only on $\Sigma$ by using a pdf of the form $$p_1(...
- 333
0
votes
1
answer
96
views
Interpolated Sobolev norm inequality
Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz set, and let $W^{k,p}(\Omega)$ denote the usual Sobolev space with $k \in \mathbb{N}$ being the order of the derivatives and $p \in [1, \infty)$...
- 13
0
votes
1
answer
210
views
Questions on the proof Lemma 4.5 GTM 175, Lickorish
I am reading GTM 175 An introduction to knot theory by Lickorish and have some questions on the proof of Lemma 4.5 given.
For (a), it says "Suppose that $C$ is amongst the $n$ components of $F\...
- 15
0
votes
1
answer
464
views
Infinite-dimensional Gaussian measure vs finite-dimensional Wiener measure
I'm trying to figure out the connections between two contructions of Gaussian measure.
Let $(U, \langle\cdot,\cdot\rangle_U)$ be a seprable Hilbert space, and $\mathcal{B}(U)$ be the Borel sigma-...
- 227
0
votes
1
answer
345
views
Embedding of fractional Sobolev space into BMO
Is it true that $$\Vert u \Vert_{BMO(\mathbb R^2)} \lesssim_{s} \Vert u \Vert_{\dot H^s(\mathbb R^2)},$$
for $s \in (0,1)$, where $\dot H^s(\mathbb R)$ is the homogeneous fractional Sobolev space?
- 99
0
votes
1
answer
241
views
Dense sub-algebra of $C_{b}((0,1))$ which is not smooth
I am looking for a dense sub-algebra $B$ in $C_{b}((0,1))$ in uniform topology such that it satisfy following requirements:
$B\cap C^{\infty}_{b}((0,1))=\mathbb{R}$ (No polynomial, no bump function).
...
- 523
0
votes
1
answer
328
views
Domain of the fractional Laplacian operator
If $u:\mathbb R^n \to \mathbb R$ satisfies $$\int_{\mathbb R^n}\int_{\mathbb R^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}} dxdy < \infty,$$
but $u$ is not in $L^2(\mathbb R^n)$, is $(-\Delta)^su$ well-...
- 109
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
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1
answer
129
views
Ordering preserved by an inverse frame homomorphism
Recall that a frame homomorphism $h:L\to M$ is called ($L$ and $M$ are frames):
Dense if, for any $x ∈ L$, $h(x) = 0$ implies $x = 0$.
Codense if, for any $x ∈ L$, $h(x) = 1$ implies $x = 1$.
...
0
votes
1
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267
views
Is this subset of $[0,1]$ of second category?
Let $S$ be an uncountable subset of $[0,1]$ such that:
$S$ is dense in $[0,1]$;
as a topological space, $S$ is Baire.
Is it true that $S$ is of second category as a subset of $[0,1]$?
- 4,459
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
324
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
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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
297
views
Mysior's example of not completely Hausdorff space
https://www.ams.org/journals/proc/1981-081-04/S0002-9939-1981-0601748-4/S0002-9939-1981-0601748-4.pdf
In this link, there is the example of regular space, that is not completely regular. This space ...
- 73
0
votes
1
answer
164
views
Restriction of non-metrizable topology to dense subset is non-metrizable
Let $(X,\tau)$ be a non-metrizable topological space which is not first-countable and let $\emptyset \neq Y\subset X$ be a proper dense subset. Is it possible for $(Y,\tau_Y)$ (where $\tau_Y$ is the ...
- 5,405
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
268
views
Topologies and Borel $\sigma$-fields on disjoint unions
Consider a set of functions $\mathcal{F}$ on $E$ where $E \subset\mathbb{R}^k$ - e.g. the class of $L_1$ functions on $[0,1]$ - and endow it with a suitable metric $d$ that makes it Polish.
Consider ...
- 195
0
votes
1
answer
554
views
Is the meaning of "irreducible manifold", "not reducible to other manifold"?
This is a cross post of MSE.
Q1: What does "irreducible manifold" mean (not definition)?
My understanding of "irreducible manifold" is "is not reducible (homotopic or ...
- 4,195
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
102
views
On a pair of continuous functions "connected" by continuous functions
Suppose $X,Y$ are topological spaces with $Y$ homogeneous and $f,g:X\to Y$ continuous such that there exist continuous functions $u,v:Y\to Y$ such that $$f = u\circ g \text{ and } g= v \circ f.$$
...
- 48.9k
0
votes
1
answer
66
views
Cardinality of the topology in countable connected $T_2$-spaces
If $(X,\tau)$ is a connected $T_2$-space with $|X|=\aleph_0$, what values can $|\tau|$ take?
- 48.9k
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