All Questions
9,780 questions
1
vote
1
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994
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On the convolution of generalized functions
It is provable that $f_\lambda\to f\Rightarrow f_\lambda*g\to f*g$ if $g$ has a compact support (shown in my textbook). In my particular case, $g=u(t+\triangle t)-u(t-\triangle t)$. Does for that ...
- 1,524
2
votes
0
answers
172
views
Projective tensor powers of Banach spaces over a normed field
Let $E$ be a Banach space over a complete normed field $\mathbb K$. Is it possible to classify all proper $E$ for which the projective seminorm $p_n$ defined on the $n$-th tensor power of $E$ is a ...
- 371
1
vote
1
answer
1k
views
Besicovitch Covering Constant for R^1
In the case where $E\subset\mathbb{R}^1$, a Besicovitch cover of $E$ is a cover by open intervals such that each point of $E$ is the center of some interval in the cover.
The Besicovitch Covering ...
- 111
0
votes
1
answer
251
views
Schrodinger Operators with diverging Potential
Is it well known that if $ H = -\bar{\Delta} + V$ (which is defined over $ L^2( \mathbb{R} ^n $ ) and $ lim_{|x| \to \infty } = + \infty $, then $ H$ has compact resolvent?
Does someone know of any ...
- 253
1
vote
1
answer
255
views
references for families of conditionaly negative definite matrices
We say that a matrix $A\in M_n(\mathbb{C})$ is a conditionaly negative definite matrix if it is hermitian and if for all complex numbers $c_1,\ldots,c_n$ such that $c_1+\cdots +c_n=0$ we have
$$
\sum_{...
- 1,222
5
votes
2
answers
579
views
Improved versions of discontinuous functions
Given a set X (such as the set of points in an interval), the space ℝX of all real-valued functions on X is not usually the function space we work with -- it is "too large" in some sense. Thus, ...
- 8,903
1
vote
0
answers
266
views
Nuclear Space problem
I need to show that if X is compact,then C(X) is nuclear.Also is the condition X is metrisable
necessary. I am at present attending a conference "Recent Aadvances in Operator Theory". This
problem ...
- 2,106
1
vote
1
answer
201
views
Reference request for sums of Grothendieck spaces
I much appreciate your help with my previous question. Now, I'd like to ask about a reference. I need a fact asserting that if $p\in (1,\infty]$ (with emphasis on the case $p=\infty$) then the $\ell_p$...
- 125
2
votes
0
answers
270
views
Spectrum of the Normal Operator associated to compact supported spectral measures
Let $\mathcal{H}$ be a Hilbert space and $E:\Sigma\to\mathcal{L}(\mathcal{H})$ be a compactly supported spectral on the Borel $\sigma$-algebra $\Sigma$ of $\mathbb{C}$. Then we can form the bounded, ...
- 229
-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
3
votes
2
answers
416
views
Which Banach spaces have categorical duals?
I was looking carefully at all the definitions, trying to understand exactly what was going on in this question on categorical duals in Banach spaces. It seems that in the category of Banach spaces ...
- 26.8k
2
votes
1
answer
536
views
about decomposition of a non-negative definite operators
Hello,
Many years before, I had the following problem.
We first give a definition. Given a non-negative definite real-valued definite matrix $n^2\times n^2$ matrix $M$, it is called separable if it ...
- 1,649
0
votes
1
answer
229
views
Weak convergence in measure for negligible sets.
Let $X$ be a Polish space and $(P_n)$ a sequence of Borel probabilities which converges weakly in measure to a Borel probability $P$. By this i mean that for any $f\in C_b(X)$ which is continuous and ...
- 125
0
votes
0
answers
362
views
Gradient of the energy functional in $H^{1,2}$-norm
I have to use estimates for the gradient of the energy functional on the free loop space of a fixed compact manifold $Q$. As such, one considers $H^{1,2}$-maps of the circle into $Q$. The energy ...
- 2,935
3
votes
1
answer
226
views
Analytic continuation of instantaneous eigenstates of a time-dependent hamiltonian
We are considering the instantaneous eigenstates of an analytically time-dependent hamiltonian and I would like to know how legitimate it is to extend them to the complex plane.
Specifically, our ...
- 2,358
6
votes
0
answers
2k
views
Weak lower semi-continuity
Which conditions assure the weak lower semicontinuity of, say, an integral functional of the type
$F(u):=\int_\Omega f(u(x),Du(x))dx$ on $W^{1,2}(\Omega,\mathbb{R}^N)$ for a bounded, if you will even ...
1
vote
2
answers
318
views
Poisson modification of subharmonic function
Let $u\in C^2(\Omega)$ be such that $\Delta u \ge 0$ on $\Omega\supset \overline{B(a,r)}$. We consider the Poisson modification $U$ of $u$ for the ball $B(a,r),$ that is $U$ equals $u$ on $\Omega-B(...
- 25
10
votes
0
answers
509
views
Lacunary hyperbolic groups and weak amenability
In the paper called Lacunary Hyperbolic group, Y. Ol'shanskii, D. Osin and M. Sapir define and characterize the lacunary hyperbolic groups, which contains the hyperbolic groups but also Tarski's ...
- 195
5
votes
0
answers
200
views
almost projective Banach space, complex scalars
It is well-known that if a real Banach space $E$ is "almost metrically projective" then $E$ is isometrically isomorphic to some $\ell^1(\Gamma)$. We say $E$ is "almost metrically projective" if ...
- 1,704
3
votes
0
answers
179
views
How to use Galerkin method to obtain existence with spaces $V \subset H$ not compactly embedded
With $V \subset H \subset V'$ a Hilbert triple (separable spaces as well), let's consider
$$u' + Au = f$$
in $L^2(0,T;V')$, where $A:V \to V'$ is bounded and linear. If $V \subset H$ is not compact, ...
- 41
0
votes
0
answers
223
views
functional equation, how to solve
Suppose $x_i, y_i \in \mathbb{R}^n$, and $F,G: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$.
$$F(x, y) = \frac{x\circ Ay}{x^TAy}$$
$$G(x, y) = \frac{x\circ By}{x^TBy}$$
where $A$ and $B$ are ...
- 13
0
votes
1
answer
365
views
Integral in a σ−convex set.
Having had no (proper) answer to this question, I formulate the remaining case as a new question as follows. With $I=[0,1]$, let $E$ be a separable (real) Banach space, and let $\gamma:I\to E$ be ...
- 3,584
1
vote
0
answers
102
views
Differentiable Path of Operators and their Inverses
Let $\mathcal{H}$ be a separable Hilbert space. Consider a differentiable map $\mathbb{R} \rightarrow \mathcal{B}(\mathcal{H}), t \mapsto A(t)$, where $\mathcal{B}(\mathcal{H})$ is the space of ...
- 11
1
vote
0
answers
289
views
Inequality regarding $\ell_p$ norms, $p<1$
Let $(x_{i,j})$ be an infinite double sequence of nonnegative real numbers, and $ 0< p<1$.
I would like to know whether one can bound from above the sum
\begin{equation}
\sum_{i,j} x_{i,j}^p
\...
- 61
0
votes
2
answers
377
views
"Frobenius-finite" linear operators on a Hilbert Space
Let $H = L_2(S)$ be the complex Hilbert space over $S$ with the counting measure. (There might be another term for this concept, but) I define a continuous linear operator $L$ on $H$ with matrix ...
user5810
0
votes
1
answer
321
views
How to handle a scalar product in an integral?
I am having a problem with a certain inequality I try to understand. I think it's just a basic idia (/trick) I'm missing, but I can't seem to find it.
Here's a simplification of the problem:
$ \...
3
votes
0
answers
1k
views
weak regularity conditions for regions to assure boundary of measure zero
Let $\Omega \subset \mathbb{R}^d$ be a region ( bounded, simply connected, open set ). What are some regularity conditions to assure the boundary $\partial\Omega$ is a set of (lebesgue-)measure zero? ...
- 131
33
votes
0
answers
1k
views
Subalgebras of von Neumann algebras
In the late 70s, Cuntz and Behncke had a paper
H. Behncke and J. Cuntz, Local Completeness of Operator Algebras, Proceedings of the American Mathematical Society, Vol. 62, No. 1 (Jan., 1977), pp. 95-...
- 25.5k
6
votes
0
answers
262
views
Given that a conditional measure is Gaussian, how bad can the original measure be?
Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration ...
- 8,512
1
vote
0
answers
52
views
Condition for maximizer of convex combination to be expansion mapping
I have $\Pi_n:\mathbb R^{n+1}\rightarrow \mathbb R$ and $F_n:\mathbb R^2\rightarrow \mathbb R$ with $$F_n(x,a)=\Pi_n(x,...,x,a)$$
$$f_n(x)=\operatorname{ArgMax}_{a\in\mathbb R}\{F_n(x,a)\} $$
such ...
- 43
1
vote
1
answer
298
views
Maximal spectrum of a complex, unital and commutative Banach-algebra
Let $A$ be a complex, unital and commutative Banach-algebra.
Question: Is the maximal spectrum $Max(A)$ of $A$ endowed with the topology induced by the prime spectrum $Spec(A)$ of $A$, Hausdorff?
...
- 328
1
vote
1
answer
304
views
How do maximum norms relatively change in Euclidean translations
Let $Q$ be the cube $[-1,1]^{3}$ and $\pi$ be a plane in $\mathbb{R}^{3}$
that contains the origin but doesn't contain any vertex of $Q$. Suppose that $A$ is an invertible
linear transformation from $\...
- 11
3
votes
0
answers
180
views
Сonvergence of the sum
This is a problem from my exam on functional analysis. I did only trivial case and now I'm just curious about another cases.
Let $T : H \rightarrow H$ is a linear continuous unitary ($T^*=T^{-1}$) ...
- 31
8
votes
0
answers
452
views
Preduals of $\ell_1$
The space $\ell_1$ has loads of (isomorphic) predulas. They can be as weird as possible but I am interested in Banach lattices.
Question: Let $X$ be a Banach lattice with dual isomorphic to $\ell_1$. ...
- 191
7
votes
1
answer
331
views
States/functionals on crossed product C*-algebras
Let $A$ be a C*-algebra, $\alpha$ a strongly continuous automorphic action by a locally compact group $G$ on $A$, and consider the crossed product $A\rtimes_\alpha G$. I am looking for references ...
- 1,017
2
votes
0
answers
128
views
How to show the identity $\int_0^T \int_{\Gamma(t)}f(s,t)\;dsdt = \int_S f(\sigma)(1+(\mathbf w \cdot \mathbf n)^2)^{-\frac{1}{2}}\;d\sigma$?
I am reading this paper.
Let $\Gamma(t)$ be a smooth closed connected oriented hypersurface for each $t \in [0,T]$. Define the set $$S = \bigcup_{t \in (0,T)}\Gamma(t) \times \{t\}.$$
On page 5 of ...
- 23
1
vote
0
answers
237
views
bivariate polynomial
Hello,
Let $p(x,y) = \sum_{m=1}^M\sum_{n=1}^N a_{m,n}x^{m-1}y^{n-1}$ be a bivariate polynomial where $\{a_{m,n}\}$ are complex.
If $(x_k, y_k), k=1,2,\cdots, MN-1$ are roots of $p(x,y)=0$ where $|...
- 9
1
vote
1
answer
630
views
Stuck on a convergence argument in $H_0^1(\Omega)$.
I'm trying to verify that a functional I have satisfies the Palais Smale condition for appliction of the Mountain Pass lemma.
However I've encountered this step along the way which seems clear to me ...
- 2,641
4
votes
1
answer
720
views
Are coordinate functionals on complete vector spaces always continuous?
(I'm just adding the completeness condition to $V$ from this 2 month old question of mine, because I realized it's relevant to whether Bill Johnson's answer to this 4 month old question of mine ...
user5810
2
votes
1
answer
134
views
Dirichlet energy with domain $W^{1,2}(M)$ or $W^{1,2}_{loc}(M)$ can be a specific Dirichlet form?
M is a Riemannian manifold, $\varepsilon(f,g)=\int_M \langle {\nabla f,\nabla g}\rangle dvol$.
Then with which domain is $\varepsilon$ a strongly local, regular and tight Dirichlet form?
$W^{1,2}(M)$ ...
- 199
1
vote
0
answers
94
views
Determining the exact form of a projection in a Hilbert space
Let $$\Omega = \left\{f(x) \in \mathcal{L}^2[0,T]: \frac{1}{T}\int_0^Tf(x)dx = \mu,~ a \le f(x) \le b,~\forall x \in [0,T]\right\},$$
where $\mathcal{L}^2[0,T]$ is the set of Lebesgue square-...
- 108
0
votes
1
answer
438
views
Möbius Transform of a Continuous Possibility Function
In order to be able to use a basic possibility function as a Body of Evidence in the Dempster-Shafer Theory of Evidence, it is needed to transform the function to its Möbius representation.
There is ...
- 103
1
vote
1
answer
295
views
A nice overview (and maybe derivation) of the Poincaré transformations of the Vector Spherical Harmonics
With $Y_{lm}(\vartheta,\varphi)$ being the Spherical Harmonics and $z_l^{(j)}(r)$ being the Spherical Bessel functions ($j=1$), Neumann functions ($j=2$) or Hankel functions ($j=3,4$) defining $$\psi_{...
- 161
1
vote
0
answers
154
views
A problem concerning measures on locally compact spaces
I am stuck on a question for quite sometime now, although in the text it is said to be "apparent". The problem goes as the following :
Let $X$ and $Y$ be locally compact Hausdorff spaces. Then $M(X)$ ...
- 111
3
votes
0
answers
209
views
A maximum of a function
When studying the $\text{UMD}$ constants of spaces like $L_{p_1}(L_{p_2}(\cdots (L_{p_n})\cdots))$, I encounter the following question: Let $\alpha > 0$, define $$C(\alpha) : = \sup_{a > 0, b>...
- 769
0
votes
1
answer
498
views
Quotient of \ell_1 by space of finite sequences
The following question came up during a reading of Rudin's functional analysis. I have not been able to find any information through searching online, but I apologise if the answer is obvious, or the ...
- 11
1
vote
0
answers
694
views
A question about an equivalent definition of the Schwartz distribution
Hello,
Does anyone know a reference or proof of the "if" part of the following statement?
$$
\mu\in \mathcal{S}'(R)\quad\text{if and only if}\quad \mu*\alpha\in\mathcal{S}(R),\forall \alpha\in C_c^\...
- 1,649
-2
votes
1
answer
295
views
When does the adjoint operator map closed convex subsets to closed convex subset?
Let $T:X\rightarrow Y$ be a linear continuous map between Banach spaces $X$ and $Y$ and denote by $T':Y'\rightarrow X'$ the norm adjoint of $T$. Let $M\subseteq U'$ be a subset of
the unit sphere $U'$ ...
- 101
3
votes
0
answers
254
views
Ways to establish equality of measures on locally compact spaces
Let $M$ be a locally compact space and $\mu$ be some probability measure on $M$. Let $y^\ast \in M$, $f(x,y)$ be a real continuous bounded function $M \times M \to \mathbb{R}$. Consider an equality
$$
...
- 1,329
0
votes
1
answer
220
views
Frames and completeness
Let $H$ be a separable Hilbert space.
A sequence $\{f_{n}\}$ is a frame for a separable Hilbert space $H$ if there exists $A,B>0$ such that for all $f$ in $H$
$$
A\|f\|^2 \leq \sum |\langle f, ...
- 71