# Tagged Questions

**0**

votes

**0**answers

125 views

### $Ax=b$ in a function space (again)

Let
$X$ be compact Hausdorff topological space,
$C(X)$ denote the algebra of complex-valued continuous functions on $X$,
$b\in \mathbb{C}^m$,
$\mathbf{A}\in C(X)^{m\times n}$,
Let ${\mathbb{C}}^n$ ...

**3**

votes

**1**answer

118 views

### $\lVert u\rVert_{W^{2,p}}$ is bounded above by $\lVert \Delta_p u\rVert_{L^2}$ for $u \in W^{1,p}_0 \cap W^{2,p}$?

I work on a bounded domain in $\mathbb{R}^n$ and let $p \geq 2$ and the operator $\Delta_p u = \nabla \cdot (|\nabla u |^{p-2}\nabla u)$.
Does the following inequality (or something similar hold) for ...

**1**

vote

**0**answers

76 views

### Free Endpoint of Minimization Problem

Consider the following minimization problem $$\inf \left\{ \int\limits_{-\infty}^0 \left[ (\psi')^2 + m(y)(\psi - F)^2 \right]\; : \; \psi \in H^1(\left(-\infty,0\right]) \right\}$$ where $m(y) > ...

**0**

votes

**0**answers

68 views

### Dipole Transition Integrals - Acceleration Form, What's Wrong?

I should have posted this question in a physics forum, but I think by posting in MathOverflow I may get more responses.
The following question may sound stupid, since I'm sure I was wrong somewhere, ...

**2**

votes

**1**answer

268 views

### Banach space of discontinuous functions(Killing continuous functions)

Edit: According to the comment of Prof. Majer, I revise the question:
For a metric space $X$, we put $A=\{f:X\to \mathbb{C}\mid \text{f is bounded}\}$. We define two semi norm on $A$
...

**2**

votes

**1**answer

210 views

### A commutative Banach algebra with an abundance of discountinuous functions

Let $A$ be the algebra of all bounded functions from $[0,\;1]$ to $\mathbb{C}$.
For $f\in A,\;$ $\omega_{f}$ is the standard oscillation function.. Each of the following two (equivalent) norms ...

**3**

votes

**1**answer

268 views

### When $C(X)$ is an injective $C(X)$-module?

It is an old question if every injective Banach space is isomorphic as Banach space to $C(X)$-space.
I would like to know if the weakened module version of this question is answered. More precisely: ...

**0**

votes

**0**answers

27 views

### continuous extensions of concave functions

Let $N$ be a lattice. For a ring R we denote $N_R := N \otimes R$. My question is the following:
Does a continuous and concave function
\begin{eqnarray*}
f: N_{\mathbb{Q}} \to \mathbb{R}
...

**3**

votes

**0**answers

92 views

### Continuously dependent on parameters [closed]

How do we check whether the solution is continuouly dependent on parameters?
Let $\Omega$ be a domain with smooth boundary. Say $f$ and $h$ are smooth. Assume that for each $\theta\in (0, 1]$, the ...

**4**

votes

**2**answers

169 views

### Can one show that the dual of a quasi-Banach space separates points without explicitly identifying the dual?

I'm interested in a question regarding the identification of some duals of quasi-Banach spaces.
However, I'm not familiar with the quasi-Banach literature, so I'm hoping somebody can point me in the ...

**0**

votes

**1**answer

150 views

### $Ax=b$ in a function space

Let
$X$ be compact Hausdorff topological space,
$C(X)$ denote the algebra of complex-valued continuous functions on $X$,
$b\in \mathbb{C}^m$,
$\mathbf{A}\in C(X)^{m\times n}$,
for all $x\in X$, ...

**2**

votes

**1**answer

109 views

### Question regarding to approximate continuity

Given $u\in BV(R^N)$, we say $u$ is approximate continuous at $x$ and the approximation limit is $l\in R$ if
$$ \lim_{r\to 0}\frac{\mathcal{L}^N(B(x,r)\cap \{|u-l|>\epsilon\})}{r^N} =0 $$
for all ...

**5**

votes

**2**answers

154 views

### Compactly supported functions and Sobolev spaces on manifolds

It is well-known that if a complete Riemannian manifold has bounded curvature and injectivity radius bounded away from zero, then the space $C^\infty_c(M)$ is dense in the Sobolev spaces $W^{k, p}(M)$ ...

**0**

votes

**0**answers

70 views

### Inverse problem to solve out current in the radiation problem

This is the radiation problem, but more like a analysis one. Suppose the current $f(x)\in L^2(\mathbb{R})$ has a compact support in $[-a,a]$. And the frequency $F(\theta)$ is given by ...

**1**

vote

**0**answers

157 views

### Separability of the space $C(C[0, 1], \mathbb{R})$

Let $E=C([0, 1])$ be the space of all real-valued continuous functions on $[0, 1]$, equipped with the uniform norm. $C(E)$ stand for the continuous real-valued functions on $E$.
I am wondering that ...

**10**

votes

**2**answers

480 views

### Banach space modulo a one-dimensional subspace =?

My question is the following:
Given an infinite dimensional Banach space $E$ and a one-dimensional linear subspace $F\subset E$. It is well-known that this one-dimensional linear subspace is closed ...

**0**

votes

**1**answer

121 views

### Equivalence of two definitions of Sobolev spaces

Good morning,
I am looking for a reference about the following fact that seems to be folklore. Define the Sobolev (Beppo Levi?) space
$$
D^{1,p}(\mathbb{R}^N) = \left\{ u \in L^{p^*}(\mathbb{R}^N) ...

**0**

votes

**0**answers

68 views

### faithful action of Hecke algebra

Let $G$ be a connected reductive group split over a number field $F$, $\mathbb{A}$ the adeles.
Let $v$ be a finite place and $\mathcal{H}_{v}$ the spherical hecke algebra at palce $v$.
...

**3**

votes

**2**answers

128 views

### Reconstructing density from integrals along specific manifolds

Let $\Phi_t : \mathbb R^n \to \mathbb R^n$ be the time-$t$-map associated to an ODE $\dot{x}=F(x)$ and let $H: \mathbb R^n \to \mathbb R$. Let $F$ and $H$ be sufficiently smooth (e.g. $C^k$ or ...

**7**

votes

**1**answer

246 views

### Bounded operator on a normed space with empty spectrum

A bounded operator acting on a complex Banach space has non-empty spectrum, and the proof of this fact uses the completeness of the space.
Is there any example of bounded operator acting on a ...

**0**

votes

**0**answers

82 views

### Defining density of a random function using Radon-Nikodym Theorem

Let $(\Omega,\mathbb{F},P)$ be a probability space and $E$ be an infinite dimensional Banach space and $\mathbb{B}$ be the $\sigma$-algebra of Borel subset of $E$.
Let $X$ be random function defined ...

**2**

votes

**0**answers

98 views

### Lebesgue point and regularity of functions

A known theorem says that for $f \in L_{loc}^1(\mathbb{R}^d)$, almost every point is a Lebesgue point.
I know too a theorem saying that for $f \in W_{loc}^{1,p}(\mathbb{R}^d)$ , every point is a ...

**5**

votes

**1**answer

141 views

### Reflexive subspaces of non-separable abstract $L_1$ spaces

An abstract $L_1$ space is a Banach lattice $E$ such that $\|x+y\|=\|x\|+\|y\|$ for disjoint $x,y\in E$. The space $L_1[0,1]$ is a separable example that contains subspaces isomorphic to $L_p[0,1]$ ...

**0**

votes

**0**answers

117 views

### Notion of solution of pde

Let's consider the following Schrodinger equation
$$iu_t+\Delta u+F(u)=0$$
in $\mathbb{R}^n$. In Cazenave's book, "Semilinear Schrodinger equation", he defines $H^1$-weak solution as $u\in ...

**17**

votes

**0**answers

252 views

### Large almost equilateral sets in finite-dimensional Banach spaces

Question: Does there exist a function $C:~(0,1)\to
(0,\infty)$ such that for each $\varepsilon\in(0,1)$ every Banach space
$X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point
set ...

**0**

votes

**0**answers

34 views

### proving that a complicated function is concave or strongly uni-modal

I am trying to prove the concave property for a complicated function during my research project (imperfect maintenance modelling for starter) which has the following form:
$\eta(t)= \alpha \beta ...

**3**

votes

**1**answer

99 views

### Example of joint cyclic and separating vector

Let $\mathcal{H}$ be a separate Hilbert space and $\mathcal{B}(\mathcal{H}) \subset
\mathcal{B}(\mathcal{H}) \otimes M_2(C)$ be a W$^*$-inclusion pairs. It is known that
this pair share a joint cyclic ...

**3**

votes

**0**answers

43 views

### Behaviour of Markov type under uniform homeomorphism of spheres

A metric space $(X,d_X)$ has Markov type $p$ (with $p \in [1,2]$), if, for every stationary Markov chain $\{Z_n\}_{n=0}^\infty$ on $Y$ (a finite space) and every mapping $f:Y \to X$, one has
$$
...

**0**

votes

**1**answer

207 views

### Continuity of a Functional

A certain functional $T$ is defined as:
$$T(F)=\int_{(0,1)}F^{-1}(s)M(ds)$$
where $M$ is a probability measure with support $[\alpha,1-\alpha]$,for $\alpha>0$.
The result that above functional is ...

**3**

votes

**0**answers

94 views

### Strong solution to $u_t - \Delta_p u = f$

For $p > 1$, consider the equation
$$\langle u_t, v \rangle + \int_\Omega |\nabla u|^{p-2}\nabla u \nabla v = \langle f, v \rangle$$
$$u(0) = u_0$$
$$u|_{\partial\Omega} =0$$
for all $v \in ...

**1**

vote

**0**answers

44 views

### decomposition of tempered distributions by entire analytic functions

Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with
$$ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: |\xi|\leq 2, \phi(\xi)=1~~\text{if}~|\xi|\leq 1\}$$
Let $j\in \mathbb N$ ...

**0**

votes

**0**answers

38 views

### Bound on change of function given bound on Hessian

Suppose I have very some smooth function $F(x)$, and let $x_0 = \text{argmin}_x F(x)$. I would like to bound $F(x) - F(x_0)$ from above, in terms of the gradient $\nabla f(x)$ and the Hessian matrix ...

**9**

votes

**3**answers

303 views

### Reference for a strong intermediate value theorem for measures

Let $\mu$ be a finite nonatomic measure on a measurable space $(X,\Sigma)$, and for simplicity assume that $\mu(X) = 1$. There is a well-known "intermediate value theorem" of Sierpiński that states ...

**3**

votes

**4**answers

310 views

### Continuity in Banach space for non-linear maps

I want to find an example of a Banach space $X$ and a continuous map $f:X\rightarrow X$ such that $f$ is not bounded on the unit ball. I do not doubt that such an example exists, but I cannot make it ...

**3**

votes

**0**answers

132 views

### Donsker's Theorem for triangular arrays

I should mention that I already posed this question on Math Stack Exchange, but didn't receive much feedback.
Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given ...

**7**

votes

**0**answers

272 views

### Proving that a space is Hilbert

Let $H=H_0^1(0,\ell)\times H_*^1(0,\ell)\times H_*^1(0,\ell)$ be equipped with the norms
\begin{align*}
...

**0**

votes

**0**answers

104 views

### A question about the duality principle

Suppose $X$ and $Y$ are finite sets and $K:X\times Y\to \mathbb R$ is some function. We get an integral transform from the space of real functions on $X$ to real functions on $Y$ given by ...

**2**

votes

**0**answers

97 views

### What's the idea behind various equivalent norms on Besov spaces $B^{s}_{p,q}$?

I am trying to understand Besov spaces; and I am eager to see why the various norms are equivalent on it.
Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with
$ \operatorname{supp} \phi ...

**2**

votes

**1**answer

156 views

### Necessary conditions for optimality in Banach spaces

Let $X$ denote the non-negative "orthant" of the Banach space $L^2$ (or whatever you call the set of functions in $L^2$ that are non-negative), and let $C$ be a closed, convex subset of $X$. Let $f$ ...

**4**

votes

**1**answer

207 views

### C*-Algebras: Dynamics vs. Derivations

Problem
Given a C*-algebra $\mathcal{A}$.
Consider dynamics $\tau:\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$ and $\tau':\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$.
(More precisely, strongly continuous ...

**1**

vote

**1**answer

161 views

### On the Hölder regularity of an integral function

Let $n\geq 3$. Let $\Omega$ be an open and bounded subset of $\mathbb{R}^n$. Let define $X_0$ as the space of functions $f:\bar\Omega\times\partial\Omega\to\mathbb{R}$ such that $f(x,\cdot)$ is ...

**0**

votes

**1**answer

254 views

### About an integral equation

I would like to obtain $g$ by solving the following integral equation
$$ \int_s^T R(u) dg(u) + f(s,T)\int_s^T g(u)du =0$$
where $f,R:\mathbb R _+ ^*\rightarrow \mathbb R _+ $and $g: \mathbb R _+
...

**4**

votes

**1**answer

217 views

### Isomorphisms between spaces of test functions and sequence spaces

I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest ...

**2**

votes

**1**answer

247 views

### Sobolev Space, “characteristic function” for the weak derivative

Let $\Omega$ be an open bounded subset of $\mathbb{R}^N$, working in the space $H_0^1(\Omega)$ with the inner product
$$(u,v)_{H_0^1} = \int_\Omega \nabla u \cdot \nabla v$$
for $u\in H_0^1$ and ...

**4**

votes

**2**answers

620 views

### What does analyticity imply in complex analysis? [closed]

In complex analysis, we're constantly faced with problems about the analyticity of a function, on which many theorems are developed. I of course know a bunch of formulas and theorems, but could not ...

**9**

votes

**1**answer

404 views

### Between compact and locally uniform: What is the name of this convergence?

Let $X$ be a topological space, $(Y,d)$ a metric space, $f\in Y^X$, and $(f_n)$ a sequence in $Y^X$ with the following property:
For every $x_0\in X$ and every $\varepsilon>0$, there exist a ...

**1**

vote

**2**answers

289 views

### How to prove that a kernel is positive definite?

For example, how to prove
$\forall(x,y)\in R^N\times R^N,K(x,y) = \displaystyle\frac{1}{1+\frac{||x - y||^2}{{\sigma}^2}}\\$
where $\sigma > 0$ is a parameter, is positive definite? I have tried to ...

**2**

votes

**1**answer

226 views

### Extensions in parabolic Hölder spaces

Let $\alpha\in ]0,1[,k\in\mathbb{N}.$Let $\Omega$ be a open and bounded subset or $\mathbb{R}^n$ of class $C^{k+\alpha}$.
As one could find in G.M. Troianello "Elliptic Differential Equations and ...

**4**

votes

**1**answer

301 views

### When is a `1-form' with continuous coefficients exact?

Let $\Omega$ be a convex, bounded open subset of $\mathbb{R}^d$, and let $C^1(\bar \Omega)$ be usual space of continuous functions on $\bar \Omega$ which are $C^1$ in $\Omega$ and whose partials in ...

**0**

votes

**1**answer

157 views

### Orthogonal projection

Let $G$ be an operator with compact resolvent on a Hilbert space $H$ such that
$\ker G \neq \{0\}$.
Further let $P$ be the orthogonal projection onto $\ker G$, and let $G_{0} := G+P$.
My question is: ...