All Questions
10,233 questions
1
vote
1
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89
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Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace - Part II
This is a follow-up to this previous question, but under stronger assumptions.
Let $(X, \mu)$ be a (say, $\sigma$-finite) measure space, let $g \in L^2$ (say, over the real
scalar field). Let $\tilde ...
- 12.6k
7
votes
2
answers
464
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Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace
Let $(X, \mu)$ be your favourite measure space (finite or $\sigma$-finite if you like), let $g \in L^2$ (say, the scalar field of $L^2$ is $\mathbb{R}$, though this probably doesn't matter). Let $\...
- 12.6k
0
votes
1
answer
124
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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
I'm reading Theorem 1 at page 98 of Vector Measures by Joseph Diestel, John Jerry Uhl. Here we use the Bochner integral.
Theorem 1 Let $(\Omega, \Sigma, \mu)$ be a $\sigma$-finite measure space, $1 \...
- 825
2
votes
1
answer
141
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Example of norm of vectors in the Tsirelson space
I would like to know if there is some expression or estimate for the norm of vectors of the form $\sum\limits_{j=1}^n e_j$ in the Tsirelson space, where $(e_j)$ is the canonical basis.
Thank you!
- 565
2
votes
1
answer
86
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Approximation with special partitions of unity
Question:
what can be recommended for calculating $f(x)$ that solves $\frac{f(x)}{f(x)+f(1-x)}\approx g(x)$ for $x\in[0,1]$?
I have tried comparing Taylor series, but they look intimidating and I ...
- 13.2k
1
vote
1
answer
260
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Finding the set of best approximation
Given $X$=$l^1$ and its dual space $X^*=l^\infty$. Now take $f=(1, 1/2, 2/3, 3/4,...) \in X^*$. Then clearly $\|f\|_\infty = 1$. I have found that $H=\ker f$ is a proximinal hyperplane in $X$.
Note: A ...
- 85
5
votes
1
answer
1k
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Left and right eigenvectors are not orthogonal
Consider a compact operator $T$ on a Hilbert space with algebraically simple eigenvalue $\lambda$. Is it then true that left (the eigenvector of the adjoint with complex-conjugate eigenvalue) and ...
- 73
9
votes
1
answer
857
views
Banach space with uncountable basis
We know that an infinite dimensional Banach space has an uncountable Hamel basis. Now if $X$ is a vector space with an uncountable Hamel basis, does there exist a norm on $X$ for which $X$ is a Banach ...
- 585
4
votes
1
answer
111
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Uniform decay of operator norm for smooth family of operators
Let $\mathscr{H}$ be a Hilbert space and let $\mathbb{R} \to B(\mathscr{H}), r \mapsto S_r$ be a continuous (or smooth) family of operators, where $B(\mathscr{H})$ is the space of bounded operators on ...
- 429
2
votes
0
answers
149
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On limits of positive linear functionals
I am looking for pointers to the literature on questions of the following kind. ($Y$ and $\Omega$ might be open subsets of some Euclidean space, but I am interested in the kind of conditions that need ...
- 3,873
0
votes
0
answers
220
views
Eigenvalue multiplicity of tensor product of positive operator with itself
Let $H$ be a separable complex Hilbert space and let $A\in B(H)$ be positive with $||A||=1$ and have eigenvalue 1 with multiplicity 1. Suppose $A=T^*T$ for some $T\in B(H)$. Denote the spectrum of $A$ ...
- 203
0
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0
answers
103
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Extension for fractional Sobolev spaces, s>0
In their paper, Fractional Sobolev extension and imbedding, the author describes all extension domains for $s \in (0,1)$ -- meaning spaces functions in which are not required to have weak derivatives. ...
- 93
4
votes
2
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191
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Reference request: "Tangent relation" in metric spaces
Let $X,Y$ be metric spaces. Let $f,g : X \to Y$ be two maps and $x_0 \in X$. Let us say that $f$ and $g$ are tangent at $x_0$ if the following condition is satisfied: For every $\epsilon > 0$ there ...
- 63.1k
1
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0
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190
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Is the strong topology the strongest?
Let $X$ be a topological vector space. We know that the weak topology $\sigma(X,X^*)$ is the weakest locally convex topology in $X$ that make every $f \in X^*$ continuous. Consequently, if $\tau$ is ...
- 203
7
votes
1
answer
738
views
Converse of closed graph theorem
Suppose $X$ is a normed linear space. If for every Banach space $Y$ and for every linear operator $T:X\to Y$, graph of $T$ is closed implies $T$ is continuous, then can we prove that $X$ is a Banach ...
- 585
1
vote
0
answers
180
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Generalizations of Fubini-Tonelli's theoem
Fubini-Tonelli's theorem: If $(X, A, \mu)$ and $(Y, B, \nu)$ are $\sigma$-finite measure spaces and $f: X\times Y \to [0,\infty]$ is a measurable function, then
$$ \int _{X}\left(\int _{Y}f(x,y)\,{\...
- 11
1
vote
1
answer
150
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Show that a certain convergence of measures is equivalent to a certain convergence of integrals
Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of a measures. We know, by the Portmanteau Theorem, that:
$$\int f d \mu_n \to \int f d\mu, \quad \forall \, f \in C_b \hbox{(class of continuous and ...
- 13
0
votes
1
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294
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Does $L_{p}(\mu, X)^*=L_{q} (\mu, X^*)$ hold for $\sigma$-finite measure spaces?
I'm reading Theorem 1 at page 98 of Vector Measures by Joseph Diestel, John Jerry Uhl.
THEOREM 1. Let $(\Omega, \Sigma, \mu)$ be a finite measure space, $1 \leq p<\infty$, and $X$ be a Banach ...
- 825
3
votes
0
answers
123
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Domain of operator which is used in operator monotone function
We are studying the paper
Rupert L. Frank, Leander Geisinger, Refined semiclassical asymptotics for fractional powers of the Laplace operator, Journal für die reine und angewandte Mathematik (Crelles ...
- 561
1
vote
1
answer
189
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Complemented subspaces in a dual Banach space
Let $Y$ be a complemented subspace in a dual Banach space $X$. Is it true that $Y$ is itself isomorphic to a dual?
This is the case of a $w^*$-closed subspace $Y$, but a complemented subspace of $X^*...
- 60.6k
0
votes
1
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232
views
A generalization about the density of $\mathcal C_c(X, E)$ in $\mathcal L_p (X, \mu, E)$ when $E$ is a Banach space
Let $X$ be a metric space, $\mu$ a $\sigma$-finite non-negative Borel measure on $X$, and $(E, |\cdot|)$ a Banach space. Let $\mathcal L_p := \mathcal L_p (X, \mu, E)$ and $\|\cdot\|_{\mathcal L_p}$ ...
- 657
4
votes
0
answers
310
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Fourier characterization of weighted Sobolev space $W^{1,2}(\mathbb R^n, \gamma_n)$
For integers $n \ge 1$ and $m \ge 0$, the Sobolev space $W^{m,2}(\mathbb R^n)$ is characterized by
$$
f \in W^{m,2}(\mathbb R^n) \text{ iff } \tilde f_m \in L^2(\mathbb R^n),
\label{1}\tag{1}
$$
where ...
- 6,853
1
vote
1
answer
264
views
Is $L_p(X, \mu, E)$ uniformly convex for $p \in (1, \infty)$ if $E$ is a uniformly convex Banach space?
Let $(X, \Sigma, \mu)$ be a $\sigma$-finite complete measure space, $(E, |\cdot|)$ a Banach space, and $p \in (1, \infty)$. Let $L_p := L_p(X, \mu, E)$ be the Bochner space of all $\mu$-integrable ...
- 657
7
votes
1
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284
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A characterization of Hilbert spaces by norm one projections
Suppose a (separable) Banach space $X$ has the following property: If $P:X\to X$ is a bounded projection different from $I$ such that $\|P\|=1$, then $\|I-P\|=1$. Does this imply that $X$ is a Hilbert ...
- 1,361
2
votes
1
answer
276
views
Construction of the Lipschitz function with a given Lipschitz constant, given two values and with small Lipschitz norm
Let the function $f\colon [a,b] \to\mathbb{C}$ be Lipschitz and let $|f(a)| \geq c,$ $|f(b)| = c$ and $\varepsilon > 0.$
It is easy to see that if $\|f\|_{\infty}< \frac{\varepsilon}{2} =: \...
- 97
3
votes
2
answers
355
views
General version of Weyl's lemma
The classical Weyl's lemma say, suppose $u \in L^1_{loc}(\Omega)$ satisfies
$$\int_{\Omega}u \Delta \phi dx=0\ \ \forall \phi\in C_c^{\infty}(\Omega),$$
then $u$ is harmonic in $\Omega.$ What I want ...
- 379
2
votes
0
answers
122
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Noether's theorem in the critical heat equation
I initially posted this question on Stackexchange Mathematics (see here) but as I got no answer, maybe someone here will be able to help me.
I am watching a serie of lectures on "Blow up solution ...
- 452
1
vote
1
answer
136
views
Construction of the Lipschitz function with a given Lipschitz constant and given two values
Let the function $f\colon [a,b] \to\mathbb{C}$ be Lipschitz and let $|f(a)| \geq c$ and $|f(b)| = c$. Is there a Lipschitz function $g$ such that $|g| \geq c,$ $g(a)=f(a),$ $ g(b)=f(b)$ and Lipschitz ...
- 97
0
votes
1
answer
161
views
Let $X^n$ be a collection of smooth functions so that their $\alpha$-Holder norms are uniformly bounded
Let $X^n$ be a collection of smooth functions so that their $\alpha$-Holder norms for $\alpha \in (1/3,1/2)$ are uniformly bounded - that is $\sup_n \|X^n\|_\alpha<\infty$. Define the standard ...
- 1,914
2
votes
0
answers
55
views
The initial sigma-algebra on the dual of a Banach lattice
Let $E$ be an AL space (i.e. a Banach lattice whose norm is additive on the positive cone $E_+$) that satisfies Mazur's condition (every sequentially weak$^*$-continuous functional on $E'$ is weak$^*$-...
- 153
3
votes
1
answer
181
views
General form of bounded linear functionals on Banach spaces
It is a famous result due to Riesz that every bounded linear functional $f$ on a Hilbert space $\mathcal{H}$ is of the form $f(x)=\langle x,z \rangle$ for a unique $z\in H$.
On p.188 of Introductory ...
- 709
7
votes
0
answers
198
views
The spectrum of the Banach algebra of certain arithmetic functions under Dirichlet convolution
I was thinking about using the tools of functional analysis to study some subring of arithmetic functions under Dirichlet convolution. If I let $D_s$ be the ring of arithmetic functions with finite ...
- 190
2
votes
1
answer
176
views
Is the measure density condition a necessary condition for bounding the Sobolev norm $W^{n,p}(\Omega)$ by the extremal terms?
Let $\Omega \subseteq \mathbb{R}^M$ be a measurable subset of positive measure. R. A. Adams and J. Fournier in their article have proven that if $\Omega$ satisfies the so-called weak cone property, ...
- 820
3
votes
1
answer
229
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A pexiderization of the sine addition law on semigroups
Can we solve the follwing functional equation
$$f(xy)=g(x)h(y)+g(y)h(x)$$
on semigroups for unknown complex valued functions $f,g,h$ ?
2
votes
1
answer
336
views
Complex Borel measures: does $\mu_n \to \mu$ weakly imply $|\mu|(\Theta) \le \liminf_n |\mu_n|(\Theta)$ for every open subset $\Theta$?
Let
$\Omega$ be a metric space,
$C_b(\Omega)$ the space of all real-valued bounded continuous functions on $\Omega$, and
$\mathcal{M}(\Omega)$ the space of all finite signed Borel measures on $\Omega$...
- 657
0
votes
1
answer
216
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Complex Borel measures: relation between the total variation norm of a measure and those of its real and imaginary parts
Let $X$ be a metric space and $\mathcal B$ its Borel $\sigma$-algebra. For $B \in \mathcal B$ we denote by $\Pi(B)$ the collection of all finite measurable partitions of $B$, i.e.,
$$
\Pi(B)=\left\{\...
- 657
2
votes
1
answer
191
views
Approximating a function by a convolution of given function?
Let $g:\mathbb{R}\to \mathbb{R}$ be a given differentiable function of exponential decay on both sides. Now let us be given a function $f:\mathbb{R}\to \mathbb{R}$, also of exponential decay, if you ...
- 20.2k
4
votes
2
answers
374
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Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation
I'm reading a proof of below theorem from this paper.
Theorem A.3. Let $\Omega$ be a locally compact normal Hausdorff space. Let $\left\{\mu_n\right\} \cup\{\mu\} \subset \mathcal{M}(\Omega)$ and ...
- 657
0
votes
0
answers
65
views
Let $E$ be Banach, $\mu_n\to\mu$ weakly on a locally compact $X$, and $f \in C_b(X, E)$. Does $\int f\mathrm d\mu_n\to\int f\mathrm d\mu$ in norm?
Let
$X$ be a metric space,
$(E, |\cdot|)$ a Banach space
$\mathcal M(X)$ the space of all finite signed Borel measures on $X$,
$\mathcal C_b(X)$ be the space of real-valued bounded continuous ...
- 825
3
votes
1
answer
608
views
On the domain of the Neumann Laplacian
Let $U$ be a bounded domain of $\mathbb{R}^d$, and write $m$ for the Lebesgue measure on $U$. For $k=1,2$, we denote by $H^k(U)$ the set of all locally $m$-integrable functions $u\colon U \to \mathbb{...
- 721
8
votes
1
answer
723
views
Bounding the discrete $l^p$ norm by the continuous $L^p$ norm for trigonometric polynomials
Let $ X_N = \text{span} \{\cos(2\pi lx): l=0, \cdots, N-1 \} $ with $ x \in [0, 1] $ and $ Y_N = \{v =(v_0, \cdots, v_{N-1}): v_j \in \mathbb{C}\} = \mathbb{C}^N $. Then $ X_N $ is the space of ...
- 105
2
votes
0
answers
136
views
Eigenfunction of $h\mapsto H(h')|_{[-1,1]}$?
Let $H$ be the Hilbert transform. Is there a continuous, even function $h:\mathbb{R}\to \mathbb{R}$ with support on $[-1,1]$ such that, for some $\lambda\in \mathbb{R}$,
$$H(h')(t) = \lambda h(t)$$
...
- 20.2k
2
votes
1
answer
455
views
A periodic integral inequality
(This problem comes in connection with a geometric problem exposed here.)
Let $\gamma(x,y)$ be a (real) function on the unit disk such that
$$
\frac{\partial^2\gamma}{\partial x \, \partial y} = 0\:\:\...
- 134
2
votes
2
answers
228
views
Minimal conditions on random vector $X \in R^n$ to ensure that $\lim_{t\to 0^+}\sup_{\|w\|_p = 1}\sup_{u \in \mathbb R}\mathbb P(|X'w-u| \le t)=0$
Let $X$ be a random variable on $\mathbb R^n$ and let $S_p^n := \{w \in \mathbb R^n \mid \|w\|_p = 1\}$ be the unit-sphere w.r.t to the $\ell_p$-norm in $\mathbb R^n$. We will be particularly ...
- 6,853
0
votes
0
answers
55
views
Weyl's law for hyperbolic operators
Let $\Omega \subset \mathbb{R}^n$ be a smooth, bounded domain and consider the operator $T: L^2([0,1]\times\Omega) \to L^2([0,1]\times\Omega)$ so that $Tf = v$ if
$$
\begin{cases}
\frac{d}{dt}v - \...
2
votes
1
answer
149
views
On a core for Neumann Laplacian on $C(\overline{D})$
Let $D \subset \mathbb{R}^d$ be a bounded $C^1$ domain. We consider a reflected Brownian motion $X=(\{X_t\}_{t \ge 0},\{P_x\}_{x \in \overline{D}})$ on $\overline{D}$. Let $\{p_t\}_{t>0}$ denote ...
- 721
0
votes
1
answer
328
views
Deduce that a function is zero on interval $[0,M]$
I have been thinking about this for the last few days but I was not able to produce a definitive answer.
Take an integrable function $g$ that maps in $\mathbb{R}$ and with domain contained in $[0,M]$ (...
- 141
1
vote
1
answer
210
views
On a property of complex exponentials
Does there exist a simple smooth closed curve $\gamma:S^1\to \mathbb C$ such that
$$ \int_{0}^{2\pi} e^{\gamma(e^{it})} \, |\gamma'(e^{it} )|\,dt =0?$$
- 4,115
6
votes
2
answers
458
views
Does the (distributional) support of the Fourier transform of an $L^p$-function with $p<\infty$ have positive measure?
Suppose that $f \in L^p(\mathbb R^n)$ such that $1\leq p < \infty$. Let $\hat f$ be the Fourier transform of $f$. Clearly, if $p=1$ or $p=2$ then the support of $\hat f$ has positive Lebesgue ...
- 437
0
votes
0
answers
246
views
Finding a unitary operator on L^{2}(\mathbb{R}^{2},dxdy)
I have a one parameter (r) family of self-adjoint representations of the universal enveloping algebra of some nilpotent Lie group on $L^{2}(\mathbb{R}^{2},dxdy)$ as follows:
$$\hat{X}^{r}=\hat{x}-i(r-...
- 103