All Questions
12,935 questions
-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 $...
- 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 ...
- 199
-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 (...
- 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
$$ \|D^k f\|_{L^{2}(0,1)} \leq \left\lfloor{\alpha\,k}\right \rfloor! \quad \forall\, k\in \mathbb N$$
...
- 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 ...
- 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|\...
- 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 ...
- 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{\...
- 433
-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 $$\eqalign{
& K:{L^2}(0,1) \to {L^2}(0,1) \cr
& f: \to (Kf)(x) = \int\limits_0^1 {k(...
- 617
-1
votes
1
answer
83
views
On probabilistic extension for Bernstein polynomials
Suppose $X_m\sim p_m(x)$ is a discrete distribution on $[0,1]$ where the value takes multipliers of $\frac{1}{m}$ (e.g., $p_m(x=\frac{k}{m})=\frac{1}{m+1})$. Suppose $p(x)=\lim\limits_{m\rightarrow\...
- 101
-1
votes
1
answer
140
views
Question to show the following function in $L^{2}$ [closed]
If $\varphi \in C^{0}(\bar{\Omega}) \cap C^{2}(\bar{\Omega} \setminus \left\{0\right\})$, does it imply that $\varphi \in L^{2}(\Omega)$?
- 19
-1
votes
1
answer
132
views
About a property in a reflexive Banach space
Let $E$ be a reflexive Banach space. Let $\{x_n\}_n$ be a bounded sequence of linearly independent elements of $E$. Does there exist a sequence $\{\phi_n\}_n$ of elements of $E^*$ (the dual of $E$) ...
- 2,118
-1
votes
1
answer
150
views
Hierarchies of Operator Norms [closed]
Given some linear operator $T: V \mapsto W$, we can talk about the operator norm between the spaces V and W, i.e.
$$
\| T \|_{V \mapsto W} \ = \ \sup_{g} \| Tg \|_W \ , \quad \mbox{ with } \| g \|_V \...
-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 $\int_{0}^{\infty}f'g dx\geq \int_{0}^{\infty}g'fdx$.
-1
votes
1
answer
59
views
Does there exist any subsequence $(u_{n_k})$ converging strongly in $L^q(\mathbb{R})$, for any $1 \le q \le \infty$? [closed]
Fix a function $\varphi \in C_c^\infty(\mathbb{R})$, $\varphi \not\equiv 0$, and set $u_n(x) = \varphi(x + n)$. Let $1 \le p \le \infty$. Does there exist any subsequence $(u_{n_k})$ converging ...
- 17
-1
votes
1
answer
226
views
separable BV space for PDE's, Whats stopping us? [closed]
Consider the metric space BV(0,1) with the following metric
$$ d(u,v) = \|u-v\|_{L^1} + |TV(u)-TV(v)| $$. It is separable, compact, uniformly bounded and complete. So What is the really obvious thing ...
- 698
-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 $\phi(x+1)=x\phi(x)$...
- 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-\...
- 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, ...
- 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 ...
- 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 ...
- 41
-1
votes
1
answer
206
views
A-priori bound on parabolic PDE that doesn't depend on end time
I have a PDE
$$u_t = a(x,t)u_{xx} + b(x,t)u_{x} + c(x,t)u + f$$
where the coefficients are in parabolic Holder space $\widetilde{C}^{0, \alpha}(I \times [0,T])$ where $I=[0,2\pi]$. The a-priori bound (...
- 1
-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 ...
-1
votes
0
answers
39
views
Linear and non-linear intersection to solve ODE
Consider a linear operator $$L(u(t)) = \dfrac{d}{dt}u(t)+p(t)u(t)$$ for known function $p(t)$. It is well known the homogeneous equation $$L(u) = 0 ~~\text{or}~~\dfrac{d}{dt}u(t)+p(t)u(t)= 0$$ has ...
-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,...
- 107
-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 ...
-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); ...
-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
- 19
-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 $A$ ...
- 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 ...
- 723
-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?
- 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
$$ \...
- 1
-1
votes
1
answer
152
views
Question regarding to the basis of L^p space via compact self adjoint operators. ( eg: inverse of -laplacian )
Do eigenfunctions of inverse of elliptic operator (eg: Laplacian) form basis of $L^P(\Omega)$ ? For p=2 we know the answer is yes, I am looking for p>2.
More generally, is it true that eigenfunctions ...
-1
votes
1
answer
696
views
Can singular measures be viewed as vanishing distributions? (Answer No!)
Hello,
Here is my original question: let $\mu$ be a singular measure with respect to the Lebesgue's measure on $R$. Is it true that $\int \psi \mu(d x)=0$ for any test function $\psi\in C_c^\infty(R)$...
- 1,649
-1
votes
1
answer
2k
views
Absolute values and Frobenius norm [closed]
The Frobenius, or Hilbert-Schmidt, norm of an $n$ by $n$ matrix $A$ is defined as $\|A\|_2 = \sqrt{\sum_{i,j=1}^n |A_{ij}|^2}$. The absolute value of $A$ is the unique positive matrix $|A|$ satisfying ...
- 1
-1
votes
1
answer
311
views
A differential equation
let $g(s)$ be real-valued function defined on $[0,T]$ such that $g(T)=0$ and suppose that $g$ is a "nice function"
Assume that $0<\gamma<1$, $v$ is a positive number, and
$$\frac{dg}{ds}+(v\...
- 1
-2
votes
1
answer
423
views
Brouwer's theorem 2.0? [closed]
Let $f\in C([0,1]^n,\mathbb R^n) $ with $[0,1]^n \subset f([0,1]^n)$
Is it true that $\exists x \in [0,1]^n, f(x) =x$?
- 4,074
-2
votes
3
answers
850
views
Books on analytic functions on Banach spaces over a non-Archimedean field
I'm looking for good textbooks on analytic functions on Banach spaces over a non-Archimedean field.
If you know one(s), please let me know.
- 1,169
-2
votes
2
answers
1k
views
Are there examples of compact infinite dimensional manifolds? [closed]
Are there known examples of compact infinite dimensional manifolds?
The word "manifold" is important.
- 73
-2
votes
1
answer
1k
views
holomorphic extension of a function [closed]
hi,
I have the following question: let $U \subset \mathbb{C}^{n}$ be some open set containing zero. let $\tilde{U} = U \cap \mathbb{R}^{n}$. assume we have a real-valued analytic function $f : \tilde{...
- 29
-2
votes
1
answer
241
views
Does a group representation being transitive on a basis imply irreducibility?
Let $G$ be an infinite discrete group and $\pi$ a representation of $G$ on the Hilbert space $H$.
Suppose that the group representation is transitive on an orthonormal basis $B = \{e_j\}_{j=1}^{\infty}...
-2
votes
1
answer
3k
views
Separability of continuous functions with compact support [closed]
Hi,
is the space $C_0(\mathbb{R}^m)$, $m \in \mathbb{N}$ of continuous functions with compact support separable? If yes: where can I find a proof for that?
Please note: this is not a duplicate of ...
- 9
-2
votes
1
answer
665
views
weak convergence
I know the following result is true in the case of strong convergence. But I don't know whether it is true in the case of weak convergence also.
Let $p>1$. Suppose that each $x_n$ is a non negative ...
- 779
-2
votes
1
answer
143
views
Relationship between noncommutative torus for different values of theta [closed]
Let $u,v\in B(L_2(\mathbb T))$ defined as $u(f)(z)=zf(z)$ and $v(f)(z)=f(ze^{-2\pi i\theta})$ for $z\in\mathbb T$ where $\theta\in\mathbb R\setminus\mathbb{Q}$. Denote the $C^*$ algebra generated by $...
- 1,879
-2
votes
1
answer
216
views
Inverse of Sobolev interpolation inequality : $\lVert u \rVert_2 \lVert \Delta u \rVert_2 \leq C\lVert \nabla u \rVert_2^2$?
If $u : \mathbb{T}^3 \to \mathbb{R}$ is a smooth function on the $3$-dimensional torus $\mathbb{T}^3$, I wonder it is possible to reverse the Sobolev interpolation inequality in the sense that
\begin{...
- 3,477
-2
votes
1
answer
1k
views
Weak convergent $+$ strongly convergent subsequence $\Rightarrow$ strong convergence? [closed]
Let $X$ be a Hilbert space containing functions defined over a bounded region $\Omega\subset \mathbb{R}^N$. Assume $f_n\in X$ converges weakly to $f\in X$, and also has a strongly convergent ...
- 395
-2
votes
1
answer
99
views
A question on the zeros involving the equation containing exponential factor [closed]
I recently encounter a puzzle that: how to show that for any constant $c_1,c_2,c_3,c_4 \in \mathbb{R}$ the equation
$$c_1 e^t+c_2e^{-t}+c_3 e^{\alpha t}+c_4 e^{-\alpha t}=0$$
has at most only one ...
- 353
-2
votes
1
answer
802
views
No Hilbert space can have countable Hamel basis without using Baire's Category theorem [closed]
I want to prove that no Hilbert space can have countable Hamel basis just using the fact that any finite dimensional subspace is closed (more specifically without using Baire's theorem). I saw a paper ...
- 317
-2
votes
1
answer
2k
views
System of first order PDE
I have a system of first-order nonlinear partial differential equations.
$$A(x,t)\frac{\partial u}{\partial t}(x,t) + B(x,t)\frac{\partial u}{\partial x}(x,t) + c(x,t) = 0$$
$$x \in \mathbb{R}, \quad ...
- 5
-2
votes
1
answer
579
views
General solution to ODE [closed]
Considering the following ODE : find $f(x)$ such that
$$\frac{\sigma^{2}}{2}\frac{d^2}{dx^2}f(x)+a(b-x)\frac{d}{dx}f(x)-(\rho+\lambda)f(x)=-\lambda g(x) $$
Where, $a,b,\rho,\lambda,\sigma\in(0,+\...
- 29