All Questions
10,198 questions
0
votes
1
answer
296
views
Continuity of cylindrical functions.
Let $C_c^\infty(\mathbb R^n)$ be the functions from $\mathbb R^n$ to $\mathbb R$ with compact support, further let $X$ be a separable Hilbert space with a fixed orthonormal basis $(e_n)_n$. Define the ...
- 455
7
votes
1
answer
537
views
Algebraic topology for nonlinear compact operators
There are analogues of certain basic notions in algebraic topology in the theory of Banach spaces. For example, the Brouwer fixed point theorem generalizes to the Schauder fixed point theorem, and ...
- 6,870
1
vote
4
answers
614
views
Variants of point fixed theorem
Let $E$ be a dual Banach space and $C$ a nonempty convex weak* compact subset of $E$. Let $G$ be a group of weak* continuous linear isometries on $E$. Suppose that $g(C)\subset C$ for all $g\in G$.
...
- 1,222
4
votes
2
answers
2k
views
Inclusions of $C^{k,\alpha}$ spaces
When is $C^{k,\alpha}(\bar{\Omega})$ a
subset of
$C^{k',\alpha'}(\bar{\Omega})$?
Gilbarg and Trudinger says that "for the domains of interest in this work the inclusion will hold whenever $k + \...
- 1,771
6
votes
2
answers
909
views
Do maps have flows?
In A New Kind of Science: Open Problems and Projects(pg. 36).
How can one extend recursive function definitions to continuous numbers? What is the continuous analog of the Ackermann function? The ...
user37691
0
votes
1
answer
319
views
Hilbert space automorphisms realized as induced by transformations of some base-spaces
Following question may be soft. Fix abstract hilbert space H and consider any automorphism A in banach-spaces sence (i.e. no conditions on metric). Call A is realizable if exist measure space $(X,\mu)$...
- 789
8
votes
1
answer
431
views
Injectivity for bimodules and Hochschild cohomology
Let $A$ be a Banach algebra and let $X$ be an $A$-bimodule. Is there a notion of (relative) injectivity for $X$ which would imply that $\mathcal{H}^n(A,X)$ vanishes for all $n\ge 1$? Here $\mathcal{H}^...
user2412
-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, $\...
- 331
6
votes
0
answers
299
views
Spectrum of an operator arising in a dynamical problem
(Question edited according to Denis Serre comment).
While studying the action of dilating map of the circle on probability measures, I ran across the following operator:
$$\mathcal{K}^* : L^2_0(\mu)\...
- 14.4k
9
votes
2
answers
1k
views
Generalization of the positive semidefinite Grothendieck inequality
In a recent paper, S. Khot and A. Naor show a natural generalization of the positive semidefinite Grothendieck's inequality. Grothendieck showed that there exists a constant $K > 0$ such that for ...
- 28.6k
2
votes
2
answers
1k
views
Are coordinate functions on topological vector spaces always continuous?
Let $V$ be a Hausdorff locally convex topological vector space over the field $\mathbb{K}$.
Let $B$ be a subset of $V$ such that
$\;$ for all functions $c : B\to \mathbb{K}$, if $\displaystyle\sum_{...
user5810
6
votes
3
answers
3k
views
Can the adjoint of the exterior derivative in semi-Riemannian geometry be defined without the Hodge * operator?
The adjoint of the exterior derivarive is defined by
$\delta:=(-1)^k\ast^{-1}d\ast$,
but I need a way which avoids the Hodge $\ast$ operator.
Is there another definition?
For example, for ...
- 4,312
0
votes
1
answer
1k
views
Linear Mapping and integration
I have been reading the paper - "Introduction to Quantum Fisher Information".
In section 1.2 the author talks about the linear map $\mathbb{J}_D$, which he defines as follows:
Let $D \in M_n$ be a ...
- 145
13
votes
6
answers
3k
views
Sets with equal positive measure in every interval
Hi,
I want to write a proof that relies on the fact that:
There are Borel Sets $A$ and $B$ contained in $\mathbb{R}$ such that
$A \cap B = \emptyset$ and $\lambda(A \cap (x,y)) = \lambda(B \cap (x,...
- 133
6
votes
4
answers
7k
views
Why do we want to have orthogonal bases in decompositions?
In the decompositions I encountered so far, we all had orthogonal set of bases. For example in Singular Value Decomposition, we had orthogonal singular right and left vectors, in [discrete] cosine ...
- 497
15
votes
2
answers
2k
views
Range of completely positive projection
Let $A$ be a C*-algebra. Suppose that $P:A \rightarrow A$ is a contractive completely positive projection. Does the range $P(A)$ is completely order isomorphic to a $C^*$-algebra?
In the case where ...
- 1,222
1
vote
2
answers
733
views
Quantum Error Correction
One can correct the errors in a quantum channel iff the coherent information of the input state is not reduced by the channel. This is analogous to sending quantum entanglement through a channel. If ...
- 29
11
votes
2
answers
2k
views
How "generalized eigenvalues" combine into producing the spectral measure?
Hi... I am wondering how 'eigenvalues' that don't lie in my Hilbert space combine into producing the spectral measure. I study probability and I am quite ignorant in the field of spectral analysis of ...
- 333
0
votes
1
answer
330
views
Convex sets and projections
Hello!
I recently started (it's purely self-education) reading a "Mathematical programming and optimizations" book, did a vast part of the exercises related to the theoretical part and at one moment ...
0
votes
2
answers
2k
views
fundamental solution of radial wave equation
i am trying to find resources on the derivation of the fundamental solution to the radial wave equation. any suggestions of or links to books, papers, and/or notes would be much appreciated. i have ...
- 1
4
votes
2
answers
1k
views
Trace space and Neumann boundary condition
In which sense is it possible to solve $\Delta u=0$, $\partial_\nu u=\phi$, for $\int\phi=0$ on a closed domain, say a ball $B^3\subset\mathbb R^3$?
For example would a $\phi\in L^p(\partial B^3)$, $...
- 2,041
4
votes
1
answer
645
views
Factorization in the Wiener algebra on the unit disc.
Consider the Banach algebra $W^+=\ell^1(\mathbb{Z}^+)$, viewed upon as the analytic functions $f$ on the unit disc $\mathbb{D}$ such that $$\|f\|=\sum_{k\ge0}|a_k|<\infty$$ where
$$f(z)=\sum a_kz^k$...
7
votes
3
answers
1k
views
Non-Borel subspace of Banach space
Let $X$ be a separable Banach space, $M \subset X$ a linear subspace. Must $M$ be a Borel set in $X$?
I believe the answer is "no," since I have seen authors who are careful to talk about "Borel ...
- 29.8k
-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
4
votes
1
answer
221
views
existence of charaterization of amenable groups by complementation?
Recall that we say that a closed space $F$ of a Banach space $E$ is complemented if there exists a contractive projection $P$ from $E$ onto $F$.
Do you know a charaterization of discrete amenable ...
- 1,222
3
votes
2
answers
1k
views
Do the Euler method's approximations always approach the true solution?
Let $B$ be a Banach space and $f : [0,+\infty)\times B \to B$ be a continuous function which is Lipschitz continuous in the second argument with Lipschitz constant $L$ (which does not depend on the ...
user5810
10
votes
4
answers
783
views
Does a quantitative version of Fredholm theory exist?
Let $X$ be a Banach space and $K:X\to X$ be a compact operator. If $I+K$ is injective then it is onto and hence the inverse $(I+K)^{-1}$ is bounded. What kind of qualitative or quantitative ...
- 9,007
2
votes
4
answers
358
views
When do functions near F have zeros near a zero of F?
Consider a sequence of functions $F_n : \mathbb{R}^d \to \mathbb{R}^d$, a function $F: \mathbb{R}^d \to \mathbb{R}^d$, and an $\mathbf{x} \in \mathbb{R}^d$ so that $F(\mathbf{x}) = \mathbf{0}$. In ...
- 1,068
4
votes
3
answers
1k
views
Set of invertible operators in B(H) is connected. Is it true? Is there a reference?
Suppose $H$ is a Hilbert space, $B(H)$ is the algebra of bounded linear operators on it, $K(H)$ is ideal of compact operators in $B(H)$, $Inv(B(H)/K(H))$ is the topological group of invertible ...
- 1,284
20
votes
8
answers
12k
views
Can a self-adjoint operator have a continuous set of eigenvalues?
This should be a trivial question for mathematicians but not for typical physicists.
I know that the spectrum of a linear operator on a Banach space splits into the so-called "point," "continuous" ...
- 355
41
votes
4
answers
16k
views
Product of Borel sigma algebras
If $X$ and $Y$ are separable metric spaces, then the Borel $\sigma$-algebra $B(X \times Y)$ of the product is the $\sigma$-algebra generated by $B(X)\times B(Y)$. I am embarrassed to admit that I ...
- 31.5k
7
votes
1
answer
577
views
Are the compact and Haagerup approximation properties equivalent?
The following essentially implies the equivalence of Anantharaman-Delaroche's compact approximation property (page 337 of Link) and the Haagerup approximation property.
Let $M$ be a type ${II}_{1}$ ...
- 7,067
6
votes
1
answer
453
views
The typical size of a random element in a Banach space
Let $X$ be a separable Banach space, and let $\mathbb P$ be a Radon probability measure on $X$ with zero mean and covariance operator $K : X^* \to X$. Let $x$ be an $X$-valued random variable with ...
- 8,512
1
vote
1
answer
210
views
Is the metric obtained by altering the metric of a Hilbert space on a finite-dimensional subspace equivalent to the original one? [closed]
Suppose a Hilbert space W can be written as the direct sum (not necessarily orthogonal) of the closed subspaces H and V, where H is assumed to be of finite dimension. Define a new inner product via
||...
- 2,935
8
votes
2
answers
8k
views
Version of the Poincaré Inequality
Let $\Omega\subset \mathbb{R}^n$ open and bounded. The Poincaré inequality
$$\|u\|_p \le C \|\nabla u\|_p$$
($\|\cdot\|_p$ denotes the usual $L^p(\Omega)$-norm; the Lebesgue measure shall be used here)...
- 2,270
2
votes
2
answers
710
views
Are there good inequalities on the norm?
It's well known that in a Hilbert space, good inequalities exist concerning the norm due to the existence of inner product.Now let X be a general Banach algebra, are there good inequalities concerning ...
- 1,528
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
3
votes
2
answers
3k
views
Eigenvalues convolution-type operator
Let $J_1$ be the Bessel function of the first kind and let $H_1(x) = \frac{J_1(|x|)}{|x|}$ for $n = 1$. Define the operator $Tf(x) = (f * H_1)(x)$ from $L^2$ to $L^2$.
Since the $H_1$-function is the ...
- 455
13
votes
1
answer
2k
views
Banach-Mazur distance between $\ell^p$-norms
Let $E^n$ be the real or complex space of dimension $n$. If $N$ and $M$ are two norms over $E^n$, and if $A$ is an endomorphism, then
$$\|A\|^M_N:=\sup_{x\ne0}\frac{M(Ax)}{N(x)}$$
is an operator norm ...
- 52.3k
0
votes
1
answer
4k
views
1
vote
1
answer
506
views
Bessel sequence, uniformly minimal, separated
Is every unit norm Bessel sequence in a Hilbert space a finite union of separated ones? Is every unit norm separated sequence a finite union of uniformly minimal (minimal with uniformly bounded ...
- 11
2
votes
1
answer
1k
views
Green's function for wave equations in R² or R³
Hello,
For almost one year, I am searching for the Green's function for wave equation in R² or R³ with some boundary conditions. As far as I know, when the boundaries permit the method of images, we ...
- 1,649
22
votes
5
answers
3k
views
Is $L^p(\mathbb{R})$ minus the zero function contractible?
Is $L^p(\mathbb{R}) \setminus 0$ contractible? My intuition says that the answer is yes, but I'm afraid that this is based on thinking of this as somehow similar to a limit of $\mathbb{R}^n \setminus ...
- 433
0
votes
3
answers
1k
views
Sobolev norm and Beppo-Levi norm
I've asked this question on math.stackexchange.com but I'm not satisfied by the answers I got, so I've decided to ask here instead. As always I apologize if my notation is not precise enough. I am a ...
- 661
2
votes
1
answer
303
views
Proper sobolev spaces invariant under no-linearities
Let $f:H^s\to H^s$ at least continuous and not necesarily linear. Is there some kind of criterion or condition over $f$ that lets to ensure that $f({H^{s+k}})\subseteq H^{s+k}$?
- 151
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
3
votes
1
answer
2k
views
fourier transform of radon measure
hi,
assume that I have a function $q$ which is a Fourier Multiplier of order zero, i.e.
$$
\left|\left( \frac{d}{dx}\right)^nq(x)\right|\lesssim \left(\frac{1}{1+|x|}\right)^n\quad \mbox{for all ...
- 979
38
votes
2
answers
13k
views
What, exactly, has Louis de Branges proved about the Riemann Hypothesis?
I know this is a dangerous topic which could attract many cranks and nutters, but:
According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] Louis ...
- 1,990
7
votes
3
answers
1k
views
Index of an Operator
I've been working through the heat-equation proof of the Atiyah-Singer index theorem. My question is what is the motivation for the definition of the index of an operator? I know there is the ...
- 205
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