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6 votes

What happens if we consider functions of bounded variation that are not in $L^1$?

The main historical reason for which the requirement $f\in L^1$ enters in the definition of $BV$ is that functions of bounded variation (tout court) of several variables were introduced by Lamberto ...
1 vote

eigenvalues of matrices (with positive entries)

This is clearly a mistake, for example the singular values of the matrix $\begin{pmatrix} 1 & 1\\1&1\end{pmatrix}$ are $(2,0)$ and not $(\sqrt 2,\sqrt 2)$. But it is harmless here, as the only ...
1 vote
Accepted

A commuting pair of isometries

Such a pair $(X,Y)$ is constructed as follows. Consider a Hilbert space $M$ with an orthonormal basis $\{e_n:n\in\mathbb Z\}$ and the bilateral shift $U$ on $M$ such that $Ue_n=e_{n+1}$. Denote by $S$...
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1 vote

Derivative of the absolute value

Here is a result that proves part 3. It is copied from the paper: P. Hajłasz, J. Maly, Approximation in Sobolev spaces of nonlinear expressions involving the gradient Ark. Mat. 40 (2002), 245--274. ...
1 vote
Accepted

Derivative of the absolute value

Part 1 follows from the Cauchy-Schwartz inequality, applied to the two vectors $(R(x), I(x))$, $(\nabla R(x), \nabla I(x))$. Part 2 follows from the simple inequality $|\partial_j R(x)| \leq \sqrt {\...
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0 votes
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Finding the set of best approximation

Similar to $P_Y(x)$, there is no such ready formula for evaluating $P_{B_Y}(x)$, when $Y=\ker (f)$, and so is for $d(x,B_Y)$. In some cases, for instance when $d(x,Y)=d(x,B_Y)$, it is easier to ...
0 votes

Uniqueness of solutions of Young differential equations

For such small $\beta$, we need to use Rough path theory to make sense of the integral and so below I go over that. (Indeed for $\beta\in (\frac{1}{2},1]$, there is an ODE theory for Young integrals ...
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0 votes

General procedure for inverse of an integral transform

I believe OP is asking: given a function $\varphi(\xi)$, can one find a function $f$ such that $\varphi(\xi) = \int_{a}^{b}f(x)g(x,\xi)dx$? This is essentially the theory of integral equations. Tables ...
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2 votes
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Reference request: "Tangent relation" in metric spaces

One reference: Elisabeth Burroni and Jacques Penon, A metric tangential calculus. Theory and Applications of Categories 23 (2010), 199–220. The first sentence of the paper says that a fuller account ...
  • 26.1k
2 votes

A ball with slit at the radius is not $W^{1,1}$-extension domain

If $\Omega$ was a $W^{1,1}$-extension domain, then restrictions of $C_c^\infty(\mathbb{R}^n)$ functions to $\Omega$ would be dense in $W^{1,1}(\Omega)$ since they are dense in $W^{1,1}(\mathbb{R}^n)$. ...
  • 1,808
4 votes
Accepted

On existence of a concave function

Such a function doesn't exist for some choices of $a$. For notational purposes I will change the unit interval by $[-2,2]$. Consider a $C^\infty$ function $a:[-2,2]\to\mathbb{R}$ such that $a(0)=3$, $...
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7 votes

What, exactly, has Louis de Branges proved about the Riemann Hypothesis?

In the years since this question was first asked and answered, there have been some new developments (namely, 80+ pages of commentary by Eric Kvaalen, and a large number of new versions of de Branges' ...
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0 votes

Is the conditional expectation of a Caratheodory function a Caratheodory function?

Here is a positive answer for the case that $\Sigma_0$ is generated by a random variable with values in a Polish space, so that we can use regular conditional probabilities and for some kernel $\kappa:...
3 votes
Accepted

Spectrum of $(Jx)_n =i((2n+1)x_{n+1}-(2n-1)x_{n-1})$ on $\ell^2(\mathbb{Z})$

Under the Fourier series isomorphism $\ell^2(\mathbb{Z}) \cong L^2(-\pi,\pi)$, $u(t) = \sum_{n\in\mathbb{Z}} x_n e^{int}$, the operator becomes $$\begin{aligned} (Ju)(t) &= 4i\sin(t) u'(t) + 2i\...
1 vote

Is the conditional expectation of a Caratheodory function a Caratheodory function?

Edit: The below answer is invalid, since $\Sigma_0$ is a sub sigma algebra of $Y$, not $X \times Y$. I think the answer is no - take $X = Y = [0, 1]$, and $f(y, x) =y$. Pick some Borel set $E \subset [...
  • 1,729
1 vote
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Understanding the Osterwalder-Schrader conditions as formulated by Glimm and Jaffe

Since this has confused me multiple times, I write this answer in the hope that it might help others. First, recall that reflection positivity as formulated by Osterwalder and Schrader states that \...
  • 539
1 vote

Example of norm of vectors in the Tsirelson space

Let $f\in W$ where $W$ is the norming set of the space and f is not a singleton. Then $f = 1/2 \sum_{j=1}^l {f_j} $ with $l\leq f_1 < ...< f_l $ all elements of $W$. Also $f$ has a ...
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8 votes
Accepted

Banach space with uncountable basis

I will pull together the comments into a community wiki answer with some of my own remarks so that the question isn't left on the unanswered questions list. If you're willing to accept that it is ...
1 vote
Accepted

Does $L_{p}(\mu, X)^*=L_{q} (\mu, X^*)$ hold for $\sigma$-finite measure spaces?

Below is my formalization of @Nik's hints to finish the proof. Let's prove that $$ \sum_{m=1}^M \|H_m\|^q_{L_{p}(\mu_m, X)^*} \le \|H\|^q_{L_{p}(\mu, X)^*} \quad \forall M \in \mathbb N^*. $$ Let $\...
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3 votes
Accepted

If $H \in L_{q} (\mu, X^*)$ such that $\int \langle H, f \rangle \mathrm d \mu = 0$ for all $f \in L_{p}(\mu, X)$, then $H=0$ $\mu$-a.e

It is always true that $L_q(\mu,X^*)\hookrightarrow L_p(\mu,X)^*$ isometrically (without the assumption that $X$ has the Radon-Nikodym property). We need the Radon-Nikodym property to guarantee that ...
2 votes
Accepted

Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace - Part II

Gro-Tsen's answer to your previous question provides a counterexample if you define $D$ to be all vectors in $\ell_2$ that are of the form $\sum_n a_n f_n$, where $f_n = e_n + e_{n+1}$, $(e_n)$ is the ...
1 vote

Can the Riemann integral be defined through a closure/completion process?

The process described in this comment actually works, and defines Riemann-Loomis integration for functions valued in Banach spaces. When the Banach space is finite-dimensional, a function is Riemann-...
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6 votes

Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace

I believe the following is a simple counterexample: Let $(X,\mu)$ be $\mathbb{N}$ with the counting measure (so I will be writing $\ell^2$ for $L^2(X,\mu)$. Let $g=0$ and $\tilde g(n) = (-1)^n$. Let ...
  • 24k
6 votes
Accepted

Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace

The answer is no and the following result provides a quite interesting counterexample. This is a known result, but I am not sure where to find it in the literature. Theorem. If $f\in L^1_{\rm loc}(\...
2 votes

Reference request: "Tangent relation" in metric spaces

I don't know whether there are "good" references to what you are actually asking but at least in some kind of implicit sense this kind of "tangency" was already considered by ...
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2 votes
Accepted

Approximation with special partitions of unity

I don't know what significance the approximate equality has, so I'll just replace $\approx$ by $=$. Any function $g(x)$ on $[0,1]$ can be split into its even $g_e(x) = (g(x)+g(1-x))/2$ and odd $g_o(x) ...
4 votes
Accepted

Left and right eigenvectors are not orthogonal

Yes, this is always true if $\lambda \not= 0$. The subsequent theorem shows a more general result. To formulate it, we need the following terminology: For an eigenvector $\lambda$ of a bounded linear ...
3 votes
Accepted

Uniform decay of operator norm for smooth family of operators

This works, essentially because $\|T^k\|^{1/k}$ for a given $k=n$ also controls this quantity for $k\ge n$. More specifically, suppose that $\|T^n\|^{1/n}\le 1-\delta$. Clearly, $\|T^{kn}\|^{1/kn}\le \...
1 vote
Accepted

On the domain of the Neumann Laplacian

This is a partial (positive) answer for the convex case only but not every detail has been worked out. Let first $U$ be convex and smooth and all functions be in $C^3$ up to the boundary. Integrating ...
2 votes
Accepted

Let $X^n$ be a collection of smooth functions so that their $\alpha$-Holder norms are uniformly bounded

This cannot hold, and in a sense rough path theory has to be developed precisely because of this reason; otherwise, rough path lifts would be defined uniquely for any curve of Hölder regularity $>1/...
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14 votes
Accepted

Converse of closed graph theorem

No. The closed graph theorem in this form is equivalent to $X$ being a barreled space. See item 15 here. There are incomplete normed spaces that are barreled. See here.
4 votes
Accepted

Show that a certain convergence of measures is equivalent to a certain convergence of integrals

It seems that this can be deduced from the Portmanteau theorem: Assume the convergence of the intergals $\int fd\mu_n$ for all $f\in C_b$ vanishing in a neighbourhood of $0$ and fix $E\in C_\mu$ with $...
3 votes
Accepted

Complemented subspaces in a dual Banach space

$L^1$ is complemented in the measure space $M([0,1])$, $L^1$ is not a dual space.
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0 votes
Accepted

A generalization about the density of $\mathcal C_c(X, E)$ in $\mathcal L_p (X, \mu, E)$ when $E$ is a Banach space

Below is a counter-example taken from this thread. It works even when $\mathcal C_c$ is replaced by $\mathcal C$, the space of all continuous functions from $X$ from $E$. Let $X:=[0, 1]$, $E:=\mathbb ...
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2 votes
Accepted

Construction of the Lipschitz function with a given Lipschitz constant, given two values and with small Lipschitz norm

I can achieve $L(f - g) \leq (\frac{1}{2} + \frac{\pi}{4})\epsilon = (1.285\ldots)\epsilon$. Two reductions: (1) we can assume $|f(t)| < c$ for all $t \in (a,b)$ and (2) we can take $\epsilon = 1$. ...
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2 votes
Accepted

Is $L_p(X, \mu, E)$ uniformly convex for $p \in (1, \infty)$ if $E$ is a uniformly convex Banach space?

A reference for this result would be Some more uniformly convex spaces by Mahlon M. Day, Bull. Amer. Math. Soc. 47(6): 504-507 (June 1941). (Alternative link at Project Euclid)
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2 votes

General version of Weyl's lemma

unfortunately Weyl lemma cannot be generalized to any divergence form elliptic operator. The problem is given by the smoothness of the coefficients $a_{ij}$. There is a huge literature on the subject. ...
2 votes
Accepted

Construction of the Lipschitz function with a given Lipschitz constant and given two values

$\newcommand\ep\varepsilon$Yes, it is easy to construct a counterexample here. Indeed, if $g_\ep$ is such a function for each given real $\ep>0$ (so that $|g_\ep| \geq c$, $g_\ep(a)=f(a)$, $g_\ep(b)...
0 votes
Accepted

Complex Borel measures: relation between the total variation norm of a measure and those of its real and imaginary parts

I represent below @NikWeaver's idea of improving $\frac{1}{2} [\cdot]' \le [\cdot]$ to get $\frac{1}{\sqrt 2} [\cdot]' \le [\cdot]$. For complex number $z = x + iy$ with $x,y \in \mathbb R$, we have $...
  • 349
4 votes

General form of bounded linear functionals on Banach spaces

For example: For the real Banach space $L^p(\mathbb R)$, with $1 < p < \infty$, the "conjugate space" is $L^q(\mathbb R)$ where $\frac{1}{p}+\frac{1}{q}=1$. For general linear ...
  • 38.5k
0 votes

Is the measure density condition a necessary condition for bounding the Sobolev norm $W^{n,p}(\Omega)$ by the extremal terms?

No, the measure density condition is not necessary I would say. Possibly there are more precise arguments, but I would argue as follows, in a nutshell: The desired inequality can be proven using ...
  • 1,808
2 votes
Accepted

Complex Borel measures: does $\mu_n \to \mu$ weakly imply $|\mu|(\Theta) \le \liminf_n |\mu_n|(\Theta)$ for every open subset $\Theta$?

$\newcommand{\Om}{\Omega}\newcommand{\Th}{\Theta}\newcommand{\B}{\mathscr B}\newcommand{\M}{\mathcal M}\newcommand\ep\varepsilon\newcommand{\de}{\delta}\newcommand{\R}{\mathbb R}$By the polar ...
2 votes

Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation

First we state here some general results about Radon measures following Bichteler, K. Integration: A Functional Approach, Birkhauser, 1991 and Bichteler, K. Integration theory: With Special Attention ...
5 votes
Accepted

A pexiderization of the sine addition law on semigroups

I corrected a typo on the right-hand-side of the equation in the OP, I'm unsure whether the left-hand-side has a typo, but the generalized sine addition law on semigroups is known in the form $$g(xy)=...
5 votes

Approximating a function by a convolution of given function?

This is an extended comment. Passing to Fourier transforms, we have the following problem: $\hat{f}$ and $\hat{g}$ are two functions analytic in some strip $-a<\mathrm{Im}\, z<a$, and bounded ...
4 votes
Accepted

Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation

$\newcommand{\Om}{\Omega}\newcommand{\Th}{\Theta}\newcommand{\B}{\mathscr B}\newcommand{\M}{\mathcal M}\newcommand\ep\varepsilon\newcommand{\de}{\delta}\newcommand{\R}{\mathbb R}$Take any $\mu\in\M(\...

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