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Definition of a $\psi$-Banach space [closed]

Let $X$ be a Banach space. Let $\mathcal{F}$ be the family of all the bounded subsets of $X$. If $I$ is the identity map on $X$, we shall denote by $\operatorname{span}\{I\}$ the vector space ...
Motaka's user avatar
  • 291
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1 answer
176 views

How do we approximate the pressure in the Boussinesq equations of hydrodynamics? [closed]

How do we approximate the pressure or the gradient of it in the Boussinesq equations of hydrodynamics ? Is the pressure limited or can it be any amount?
mahdi's user avatar
  • 11
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1 answer
70 views

Is this kind of interpolation correct?

Let $f=\sum f_j$ be a finite sum. Assume that $$ \|f\|_2\le(\sum\|f_j\|_2^2)^\frac12$$ $$\|f\|_\infty\le C\max_j\|f_j\|_\infty$$ Then can we conclude that for $2<p<\infty$ $$\|f\|_p\le C^{1-\...
xsbb2001's user avatar
-1 votes
1 answer
328 views

About the critical points of quasi-convex functions

What do we know about the structure of critical points of quasi-convex functions? I am looking for statements like "the critical points of a quasi-convex function are always either a global minima ...
gradstudent's user avatar
  • 2,246
-1 votes
1 answer
346 views

An infinite set in a compact space

Let $X$ be a topological space. Is there any characterization for the property that says "for every infinit subset $A$ of $X$ there exists $a\in A$ such that if $f$ be an arbitrary real continuous ...
robert caro's user avatar
-1 votes
1 answer
360 views

Which operators other than self-adjoint operators have no purely imaginary eigenvalues? [closed]

Given an operator mapping between suitable spaces, what is the condition that guarantees all eigenvalues have nonzero real part? Obviously self-adjointness implies all eigenvalues are real, but how ...
Tobi's user avatar
  • 7
-1 votes
1 answer
148 views

Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$ [closed]

Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology. Let $X$ be a topological space (for convenience, it might be Polish ...
user66910's user avatar
-1 votes
1 answer
104 views

Question about measure lemma?

"Let (u_j) be a bounded sequence from $W^{1,p}(\Omega)$ how to prove that there exists a subsequence such that $u_j\rightharpoonup u$ in $W^{1,p}_0(\Omega)$ and $|\nabla u_j|\rightharpoonup d\mu,$ $|...
Vrouvrou's user avatar
  • 277
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1 answer
516 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) \...
Paperino's user avatar
-1 votes
1 answer
128 views

Proving convergence of an integral-differential equation [closed]

I have a second order nonlinear ordinary differential equation which I transformed into an integral-differential equation by multiplying the ODE by $y'$ and integrating. My question is where can I ...
Alan's user avatar
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-1 votes
1 answer
187 views

Limit of a function in a weighted Sobolev space

I have a function $f(x)$ in the space $H^{2,-s}(\mathbb{R}^3)$; have this limit sense $$\lim_{|x-y|\to 0} f(x)$$ ? ($y$ is a fixed point) If i have $f$ in $H^2$ I can say that $$\lim_{|x-y|\to 0} f(x)=...
Sue's user avatar
  • 25
-1 votes
1 answer
1k views

relation between inclusion and embedding [closed]

Assume that $X$ and $Y$ are two Banach spaces, now we have that $X$ is included in $Y$, in the sense that $\forall a\in X$, we have $a\in Y$. Then can we get that $X$ is embedded in $Y$, namely, $\...
Shaoming Guo's user avatar
-1 votes
1 answer
286 views

Check an equation on the Heisenberg group $H_1$

The Heisenberg group $H_1$ is the set $\mathbb C\times \mathbb R$ endowed with the group law $$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right); \quad \forall z,w \in \mathbb C\,...
Z. Alfata's user avatar
  • 650
-1 votes
1 answer
153 views

Sobolev estimates $\|\nabla\phi\|_{\infty}\leq C\|\phi\|_{H^2}$

This is a cross post in continuation to this question on Mathematics Stack Exchange. I wanted to know if this inequality holds true in two or three dimensions, $\|\nabla\phi\|_{L^{\infty}(\Omega)}\leq ...
Mainak's user avatar
  • 101
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1 answer
79 views

A question about the commutator $[J^s,u]\partial_x u$

I am studying the use of the commutator for finding the estimate of energy. During my looking through many papers I found that this paper contains a possible typo. Here is the archive version which ...
Mr. Proof's user avatar
  • 159
-1 votes
1 answer
164 views

Closure of the point spectrum of an unbounded diagonalizable operator

Given a (separable) Hilbert space H and an unbounded densely defined linear operator $T:{\cal D}(T) \to $H such that ${\cal D}$ is diagonalizable (it means $\exists$ an O.N.B. of H such that all basis ...
Dave Shulman's user avatar
-1 votes
1 answer
78 views

Fundamental of a signal

Consider the space $S$ of real functions with the norm $$\|f\|^2 = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} e^{-x^2/2} f^2(x) ~\mathrm{d}x, $$ or any reasonable Euclidean norm such that bounded ...
Arthur B's user avatar
  • 1,902
-1 votes
1 answer
210 views

A commuting pair of isometries

Let $H$ be a Hilbert space and $B(H)$ be the space of all bounded operators on $H$. The Wold decomposition says that: an operator $x$ in $B(H)$ is an isometry if and only if $x=x_u\oplus x_s$ where $...
ABB's user avatar
  • 4,058
-1 votes
1 answer
320 views

Existence of weak limit for bouded sequence $\{y_n\}$ such that for every weak limit point $\{y_n\}$ must equal $y$

Let $X$ be separable Banach space and $\{x_n\}$ be a bounded sequence, relatively weakly compact sequence in $X$. we set $y_n=\frac{1}{n}\sum_{i=1}^{n}{x_i}$, then (together with the Krein and ...
Karim KHAN's user avatar
-1 votes
1 answer
323 views

Expressing the sum of two squared inner products more compactly: is it possible to lift the dimension? [closed]

Let $v_1,v_2\in\mathbb{R}^d$ be two fixed vectors, and $\langle \cdot,\cdot\rangle_{\mathbb{R}^d}$ be the usual Euclidean inner product in $\mathbb{R}^d$. My question is as follows. Is there an (...
hookah's user avatar
  • 1,096
-1 votes
1 answer
119 views

Existence of a function with slow growth on derivatives

Does there exist a smooth compactly supported function $$f \in C^{\infty}_c((0,1))$$ such that $$ \|D^k f\|_{L^{2}(0,1)} \leq \left\lfloor{\alpha\,k}\right \rfloor! \quad \forall\, k\in \mathbb N$$ ...
Ali's user avatar
  • 4,143
-1 votes
1 answer
81 views

Closed on generic set implies closed set whole set [closed]

Assume that $f:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ is a continuous on a set $A$. Let $B \subset A$ be a generic set in $\mathbb{R}^{2}$ i.e, the countable intersection of the open and dense ...
Adam's user avatar
  • 1,043
-1 votes
1 answer
349 views

Sequence converging to different limits with respect to two different _complete_ norms

Do there exist a real vector space $X$ complete with respect to norms $|\cdot|$ and $\|\cdot\|$ and a sequence $(x_n)_{n\in \mathbb N} \subset X$ such that there exist $x,y\in X$: $x\ne y$, $|x_n - x|\...
Skeeve's user avatar
  • 1,277
-1 votes
1 answer
74 views

Invariant ergodic measure Volterra operator

Define the Volterra operator $V$ on $C_0([0,1])\triangleq \{g \in C([0,1]):g(0)=0\}$ by $$ f \mapsto \int_0^{\cdot} f(s)ds. $$ Is there an example of an ergodic and $V$-invariant Borel probability ...
ABIM's user avatar
  • 5,405
-1 votes
1 answer
265 views

A sequence of Hilbert spaces and a sequence of linear funtionals [closed]

Let $H$ be an Hilbert space over $\mathbb{C}$ Let $\{h_m\}_{m \in \mathbb{N}} \subset H$ be a sequence of linearly independent vectors in $H$ Let $\forall m \in \mathbb{N}: H_m = \overline{\...
Matey Math's user avatar
-1 votes
1 answer
102 views

Compactness of a special kind of Integral operators

Let $(S(t))_{t>0}$ be a continuous operator from $L^2(0,1)$ to its self and Let $K$ be the operator $$\eqalign{ & K:{L^2}(0,1) \to {L^2}(0,1) \cr & f: \to (Kf)(x) = \int\limits_0^1 {k(...
Gustave's user avatar
  • 617
-1 votes
1 answer
83 views

On probabilistic extension for Bernstein polynomials

Suppose $X_m\sim p_m(x)$ is a discrete distribution on $[0,1]$ where the value takes multipliers of $\frac{1}{m}$ (e.g., $p_m(x=\frac{k}{m})=\frac{1}{m+1})$. Suppose $p(x)=\lim\limits_{m\rightarrow\...
ZUN LI's user avatar
  • 101
-1 votes
1 answer
140 views

Question to show the following function in $L^{2}$ [closed]

If $\varphi \in C^{0}(\bar{\Omega}) \cap C^{2}(\bar{\Omega} \setminus \left\{0\right\})$, does it imply that $\varphi \in L^{2}(\Omega)$?
dyyyyssss's user avatar
-1 votes
1 answer
132 views

About a property in a reflexive Banach space

Let $E$ be a reflexive Banach space. Let $\{x_n\}_n$ be a bounded sequence of linearly independent elements of $E$. Does there exist a sequence $\{\phi_n\}_n$ of elements of $E^*$ (the dual of $E$) ...
MSMalekan's user avatar
  • 2,118
-1 votes
1 answer
150 views

Hierarchies of Operator Norms [closed]

Given some linear operator $T: V \mapsto W$, we can talk about the operator norm between the spaces V and W, i.e. $$ \| T \|_{V \mapsto W} \ = \ \sup_{g} \| Tg \|_W \ , \quad \mbox{ with } \| g \|_V \...
Atransportconfusion's user avatar
-1 votes
1 answer
136 views

An elementary question about integration by parts! [closed]

Let $f,g: R \rightarrow R$ be two positive increasing functions. Under what (non-trivial) conditions one can guarantee that $\int_{0}^{\infty}f'g dx\geq \int_{0}^{\infty}g'fdx$.
A random mathematician's user avatar
-1 votes
1 answer
173 views

finding a unitary submatrix inside a random matrix

Let $\mathbf{R} \in \mathbb{C}^{~m \times n} $ with $m \leq n $ be a random matrix, whose entries are i.i.d zero mean random variables with circularly symmetric Normal distribution. Let where $r$ be ...
Jeff's user avatar
  • 482
-1 votes
2 answers
187 views

On Bohr-MollerupTheorem [closed]

In http://mathworld.wolfram.com/Bohr-MollerupTheorem.html, Bohr-Mollerup Theorem is given where it is stated that $\Gamma$ function is the unique log convex function that satisfies $\phi(x+1)=x\phi(x)$...
Turbo's user avatar
  • 13.9k
-1 votes
1 answer
159 views

Question about the derivative of a fuctional

I have this lemma+proof and i dont understand why it follows from $J'(u_n)\rightarrow 0$ that $-\Delta_p u_n- f(x,u_n)\rightarrow 0$ such that $J(u)=\frac1p\int_{\Omega} |\bigtriangledown u|^p dx-\...
Vrouvrou's user avatar
  • 277
-1 votes
1 answer
75 views

Finiteness of "novel variance" from a kernel on a compact space [closed]

Let $c(i,i')$ be a kernel function on a reasonable index space $I$. Choose a dense sequence of points $\{i_1, i_2, \cdots \} \subseteq I$, and define the one-point kernel functions $k_n := c(\cdot, ...
Tom LaGatta's user avatar
  • 8,512
-1 votes
1 answer
211 views

Stone Cech compactification for exponential map

Recently I met with a problem related to Stone-Cech Compactification theorem in Furstenberg's famous paper "non-commuting product." I try my best to understand Stone-Cech compactification theorem by ...
yaoxiao's user avatar
  • 1,706
-1 votes
1 answer
259 views

Absolute continuity of probabilities on Polish spaces and open sets. [closed]

On a polish space $\mathcal{X}$ i consider two Borel probabilities $P$ and $Q$ such that for any open set $E$ of $\mathcal{X}$ we have : $P(E) =0$ implies $Q(E)=0$. Does this imply that $Q$ is ...
Polite's user avatar
  • 41
-1 votes
1 answer
934 views

Domain and exponential of self- adjoint operator

Let $A$ be a self - adjoint operator on a Hilbert space $\mathcal{H}$ and let $D(A)$ be its domain. If $\psi \in D(A)$ then $exp(-itA) \psi \in D(A)$ iff $A$ is bounded ? Thank ...
Physics  beginner's user avatar
-1 votes
0 answers
94 views

Why define Schwartz by supremum rather than limit?

The Schwartz space is defined as the set of all indefinitely differentiable functions such that the supremum over the free variable of any (order) derivative times any (order) power is finite. However,...
Ponder Stibbons's user avatar
-1 votes
0 answers
53 views

convergence of convolution in Bochner space

I want to prove a well-known fact in $L^p(R^n)$ namely that, the convolution of an element in $L^p$ with an element of $L^1$ is in $L^p$ let: if $u∈L^p (R;X) , f∈L^1 (R)$ and $X$ is Separable and ...
Alucard-o Ming's user avatar
-1 votes
1 answer
86 views

how take weak derivative of norms in hilbert spaces?

Let the following be hilbert spaces with dens inclusions $V ↪H=H^* ↪V^*$. Where $H^*$ and $V^*$ are the duals. $H$ has the product $(*,*)$ and $V×V^*$ has the product $⟨*,*⟩$. Let $u∈L^2 ([0,T];V); ...
Alucard-o Ming's user avatar
-1 votes
1 answer
118 views

Sobolev injections [closed]

It is true to write that $W^{1,\infty}(]0,\infty[) \hookrightarrow C([0,\infty[)$ et $W^{1,1}(]0,\infty[) \hookrightarrow C([0,\infty[)$ ? Thanks
user895874's user avatar
-1 votes
1 answer
246 views

Determine the singular values of a compact operator in terms of the eigenvalues of an alternating tensor product of operators

Let $H$ be a $\mathbb R$-Hilbert space, $A\in\mathfrak L(H)$ be compact and $$|A|:=\sqrt{A^\ast A}$$ denote the square-root of $A$. By definition, the $k$th largest singular value $\sigma_k(A)$ of $A$ ...
0xbadf00d's user avatar
  • 167
-1 votes
1 answer
114 views

Construct a probability function on the operator monotone functions, $g(t)=t g(t^{-1})$, fitting certain values

To immediately pose the question of interest to us, without first expanding upon its (quantum-information-theoretic) origin—we seek a univariate function $f$, for which we have the ("two-qubit ...
Paul B. Slater's user avatar
-1 votes
1 answer
77 views

Parseval frame, convergence of $\sum_{k=0}^\infty \left\|g_k\right\|$ [closed]

Let $\mu$ be a Borel probability measure on $[0, 1)$, and $\{g_k\}_{k=0}^\infty$ be a Parseval frame for $L^2(\mu)$. Does $$\sum_{k=0}^\infty \left\|g_k\right\|$$ converges?
Mark's user avatar
  • 297
-1 votes
2 answers
440 views

Motivation for weak solution of a PDE (initial condition)

The following question came to me when reading the famous paper of ALT and LUCKHAUS: "Quasilinear elliptic-parabolic differential equations" When looking at a (nonlinear degenerate) PDE like $$ \...
Trant34's user avatar
-1 votes
1 answer
293 views

spectrum of a special class of tridiagonal matrices

Consider a real and symmetric tridiagonal matrix with zero diagonals and where subdiagonals and superdiagonals are equal to 1 except the (1,2)-th component being equal to $a$, i.e., $$\begin{bmatrix}...
Tanyanat's user avatar
-1 votes
1 answer
152 views

Question regarding to the basis of L^p space via compact self adjoint operators. ( eg: inverse of -laplacian )

Do eigenfunctions of inverse of elliptic operator (eg: Laplacian) form basis of $L^P(\Omega)$ ? For p=2 we know the answer is yes, I am looking for p>2. More generally, is it true that eigenfunctions ...
user45267's user avatar
-1 votes
1 answer
696 views

Can singular measures be viewed as vanishing distributions? (Answer No!)

Hello, Here is my original question: let $\mu$ be a singular measure with respect to the Lebesgue's measure on $R$. Is it true that $\int \psi \mu(d x)=0$ for any test function $\psi\in C_c^\infty(R)$...
Anand's user avatar
  • 1,649
-1 votes
1 answer
2k views

Absolute values and Frobenius norm [closed]

The Frobenius, or Hilbert-Schmidt, norm of an $n$ by $n$ matrix $A$ is defined as $\|A\|_2 = \sqrt{\sum_{i,j=1}^n |A_{ij}|^2}$. The absolute value of $A$ is the unique positive matrix $|A|$ satisfying ...
Chris's user avatar
  • 1