**0**

votes

**0**answers

30 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 ...

**2**

votes

**0**answers

42 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

74 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 ...

**13**

votes

**0**answers

134 views

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

Question: Does there exist a function $C:~(0,1)\to
\mathbb{R}$ 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

25 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 ...

**2**

votes

**0**answers

25 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

40 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
$$
...

**-2**

votes

**0**answers

45 views

### Weak topology and topology by semi-norm [on hold]

Wikipédia:
-The weak topology on X is the initial topology with respect to X* (let's note it T')
-If the field K has an absolute value , then the weak topology σ(X,F) is induced by the family of ...

**0**

votes

**1**answer

178 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 ...

**2**

votes

**0**answers

68 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**

votes

**0**answers

55 views

### Continuous versions of tensors/ Tensors with infinite indices?

In linear algebra and general relativity, we knew that vectors can be represented by a linear combination of components and a basis
$$\mathbf{V}=\sum_{i=1}^n A_i\mathbf{e_i}$$
Or in Einstein ...

**1**

vote

**0**answers

36 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

34 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 ...

**8**

votes

**3**answers

268 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 ...

**-2**

votes

**0**answers

53 views

### An analytic characterization of eigenvalues of a Hermitian matrix [on hold]

[..the following is trying to understand a certain argument of Terence Tao in a lecture notes of his..]
If $A$ is a Hermitian matrix with eigenvalues, eigenvectors $\{ \lambda_i , e_i \}_{i=1}^n$. ...

**3**

votes

**4**answers

264 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

119 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 ...

**-4**

votes

**0**answers

63 views

### About diagonal entries of the graph Laplacian [on hold]

[..in the following you can assume its a regular graph if necessary..]
Is anything special known about them?
Are they characterized in any other way?
Is the largest diagonal entry in any power of ...

**7**

votes

**0**answers

252 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

96 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

81 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 ...

**0**

votes

**0**answers

35 views

### What is the Vapnik-Chervonenkis dimension of sigmoidal functions? [migrated]

Consider the following class of functions: $F=\{f_w:R^d \rightarrow [a,b], f_w(x)=\sigma(w^Tx), \forall x\in R^d\}$, where $\sigma(\cdot)$ is a sigmoidal function (e.g. tanh, or sigmoid so it has ...

**0**

votes

**0**answers

105 views

### Unitary operators on Hilbert spaces [closed]

Consider the set $U(H)$, of unitary operators on the real separable Hilbert space $H$. Fix an orthonormal basis $\{e_i\}$ of $H$. Is the subset $S$ of $U(H)$ corresponding to finite dimensional ...

**2**

votes

**1**answer

84 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 $f$ ...

**3**

votes

**1**answer

195 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

154 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

248 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

173 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

221 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

599 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

342 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

276 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

212 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

294 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

151 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: ...

**7**

votes

**7**answers

547 views

### What is the best reference for Spectral theory?

I'm studying Bernard Aupetit: A Primer on Spectral Theory
but the textbook we are using is a little bit heavy going for me. Is there a best book to learn about these things?
Thank you.

**0**

votes

**0**answers

48 views

### A question related to the Nikolskii fractional spaces

Consider a Nikolskii space, that is
$$
N^{s,p}=\{f\in L^{p}(I, d\ell): \|f\|_{\overline{N}^{s,p}}=\underset{h>0}{\sup}h^{-s}\|\tau_{h}f-f\|_{L^{p}(I_{h})}<\infty \},
$$
where ...

**6**

votes

**0**answers

137 views

### Is an infinite-dimensional “Lebesgue measure” uniquely determined by a set of positive finite measure?

Let $\mu$ be a probability measure on a subset $C \subset \mathbb{R}^\infty$ of the space of sequences, and assume, for simplicity, that $C$ is closed and convex.
We say that $\mu$ admits shifts if ...

**1**

vote

**0**answers

81 views

### continuity with respect to weak-${\ast}$ topology

Let $V:=V([0,1],R)$ be the space of all cadlag functions defined on $[0,1]$ of bounded variation. Thus any element $v\in V$ determines a signed measure $\nu$ on $[0, 1]$ given by the formula $\nu([0, ...

**-4**

votes

**2**answers

97 views

### Does the Laplacian commutes with the indicator function [closed]

We define the laplacian operator $\Delta$ with the Neumann boundary conditions on the space $H^2(\Omega)$, where $\Omega$ is an open set of $\mathbb{R}^n$ with a smooth boundary $\partial\Omega$, and ...

**3**

votes

**1**answer

100 views

### A question about Skorokhod metric

I have a question related to the Skorokhod distance.
Let $\Omega:=D([0,1],R)$ be the space of cadlag functions $x$ defined on $[0,1]$. Let $\Lambda$ be the collection of non-decreasing continuous ...

**1**

vote

**1**answer

134 views

### Smooth curves in a Frechet space

Is the space $C^{\infty}([0,1];C^{\infty}(S^1))$ equal with the space $C^{\infty}([0,1]\times S^1)$ ? I am interested in characterizing the smooth curves in the space $C^{\infty}(S^1)$ where $S^1$ is ...

**3**

votes

**1**answer

74 views

### Skorokhod distance between $\omega, \omega\circ f_{\varepsilon}$ and $\omega, \omega\circ b_{\varepsilon}$

Let $\Omega:=D([0,1],R)$ be the space of cadlag functions $x$ defined on $[0,1]$. Let $\rho$ be the Skorokhod metric on $\Omega$, see e.g.
http://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g
Now define ...

**3**

votes

**1**answer

139 views

### A nilpotency question on $C^{*}$ algebras

What is an example of a $C^{*}$ algebra $A$ with the property that: for every nilpotent(Quasi nilpotent) $a$ and for every $n\in \mathbb{N}$, there is a $b$ with $b^{n}=a$.
To what extent ...

**6**

votes

**0**answers

156 views

### What function space does holomorphic functional calculus give us?

Let $A$ be a unital Banach algebra, $U$ be an open subset of $\mathbb{C}$, and $A_U:=\{x\in A:\sigma(x)\subset U\}$. Holomorphic functional calculus says that any holomorphic function ...

**4**

votes

**1**answer

81 views

### Approximation of sets by sets with regular border

What kind of conditions on a (bounded) set $E \subset \mathbb{R}^{n}$ ensure that it can be approximated from outside/inside by sets with regular border (say Lipshitz or $C^{k}$ conditions) in the ...

**2**

votes

**0**answers

89 views

### Counterexample for closed graph theorem in unmetrizable case

Consider locally convex vector spaces $X,Y$ and let $A: X\to Y$ be a linear operator with closed graph. Closed Graph Theorem states that if $Y$ is Fréchet then $A$ is continuous.
What is a ...

**1**

vote

**0**answers

39 views

### Representing a Pullback as an Infinite Matrix

Let $M$ and $N$ be manifolds and let $T: M \to N$ be a bijective map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ \mathcal{F}(N,\mathbb{R})$) be the space of all functions from $M$ (resp. $N$) to ...

**1**

vote

**0**answers

91 views

### Properties of a function from its pullback

Edit: I have now removed the duplication previously referred to. Thank you.
Let $M$ and $N$ be smooth manifolds and $T: M \to N$ be a smooth map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ ...

**2**

votes

**1**answer

123 views

### Inverse Problem for Pullback

Let $M$ and $N$ be smooth manifolds and $T: M \to N$ be a smooth map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ \mathcal{F}(N,\mathbb{R})$) denote the space of smooth functions from $M$ (resp. $N$) ...