All Questions
1,154 questions
10
votes
3
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Pathological product space norm
Let $X$ and $Y$ be two normed vector spaces and $n(\cdot, \cdot)$ be any norm on $\mathbb{R}^2$. Is it always possible to define a norm on the product vector space $X \times Y$ as $||(x, y)||_{X \...
- 202
10
votes
1
answer
1k
views
Separating vectors for C$^*$-algebras
(I asked this on math.stackexchange, without response).
Let $A$ be a C$^*$-algebra, concretely acting on a Hilbert space $H$. Suppose that $\xi_0\in H$ is cyclic and separating for $A$ (that is, the ...
- 18.7k
10
votes
1
answer
658
views
Are functions of moderate growth a bornological space?
I was thinking a bit about distribution theory the last weeks and stumbled across the following question:
There are two natural locally convex topologies on the space of smooth functions of moderate ...
- 9,650
10
votes
4
answers
1k
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References: Infinite dimensional Lie algebras
What I really want are properties (if it is abelian, nilpotent, solvable, simple, or semisimple; Cartan subalgebras...) of the Lie algebra of smooth functions on a symplectic manifold $(M,\omega)$; ...
- 641
10
votes
1
answer
700
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Reference request: Riesz potential $I_\alpha : L^{d/\alpha} \to \rm{BMO}$?
Let us denote the Riesz potential in $\mathbb R^d$ by
$$
I_\alpha (f)(x) := c_{d, \alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}}
\, dy.$$
By the classical Hardy-Littlewood-Sobolev theorem ...
- 632
10
votes
2
answers
1k
views
Reference Request: Probability and (Nonlinear) PDEs
I'm a graduate student interested in learning about probability and (mostly evolutionary) PDEs, just for fun (and as an excuse to learn some probability). I'm mostly interested in things along the ...
- 101
10
votes
1
answer
2k
views
Counting norms on an infinite dimensional vector space
It is known that whenever E is a finite dimensional real vector space, there is only one norm on E up to equivalence (actually one non discrete vector space topology).
Is it known what happens when E ...
- 101
10
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1
answer
598
views
What happens if we rotate the kernel of an integral operator?
Given an integral operator $K$ on $L^2(\mathbb R)$ with kernel $k(x, y)$, consider the integral operator $L$ on $L^2(\mathbb R)$, whose kernel has the form $k(\alpha x+\beta y, \gamma x+\delta y)$, ...
- 452
9
votes
1
answer
621
views
Uniqueness of solutions of Young differential equations
Consider the following one dimensional Young differential equation:
\begin{align*}
&Y_t=\int_0^t Y_s dX_s,\quad t\in[0,1];\\
&Y_0=0.
\end{align*}
Here the driving process $X$ is a bounded ...
- 931
9
votes
3
answers
1k
views
Does there exist a notion of discrete riemannian metric on graph?
I would like to know if there is any notion of a discrete Riemannian metric on graphs. C. Mercat has worked on discrete Riemann Surfaces, but that's not exactly what I am working on.
To be more ...
- 91
9
votes
2
answers
553
views
Asymptotic behavior of Sturm-Liouville eigenvalues
I have two questions.
Consider the operator $Av = -v'' + a(x)v$ on $I = (0, L)$, with zero Dirichlet condition and $a \in C([0, L])$.
Let $(\lambda_n)$ denote the sequence of eigenvalues of $A$....
- 369
9
votes
1
answer
596
views
Why is the Berkovich spectrum of a C*-Algebra the same as the Gelfand spectrum?
Let $A = \mathcal{C}(X)$ be a commutative (unital) C*-Algebra. Let $Spec(A)$ denote its Gelfand spectrum
$$ Spec(A) = \{A \rightarrow \mathbb{C} : \text{non-zero *-homomorphism} \} \simeq X. $$
Now ...
- 335
9
votes
1
answer
3k
views
Is every finite Borel measure on a locally compact Hausdorff, $\sigma$-compact and separable space automatically regular?
The conditions stated in the question seem mouthful and a bit arbitrary, so let me provide some backgrounds.
Definition
Let $\mu$ be a Borel measure on a topological space. We say:
$\...
- 83
9
votes
1
answer
652
views
Scaling in Mehta's integral
The following expression is known as Mehta's integral and deeply connected to random matrix theory:
$$\frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \prod_{i=1}^n e^{-...
- 124
9
votes
2
answers
758
views
Number of critical points of smooth functions on $S^1$
Let $u$ be a smooth function on the unit circle $S^1$ such that $\int_{S^1}ux_j=0$, for $j=1,2$. Is the number of critical points of $u$ strictly bigger than 2?
9
votes
1
answer
758
views
Convergence of Schwartz kernels implies convergence of operators
Let $K$ be a smoothing operator on $\mathbb{R}^n$, i.e., it defines a map on all Sobolev spaces $K\colon H^r(\mathbb{R}^n) \to H^s(\mathbb{R}^n)$ for all $r, s \in \mathbb{R}$. Now (a variation of) ...
- 2,998
9
votes
1
answer
499
views
Subspaces of $L^2(0,1)$ dense on every truncation $L^2(c,1)$
It may be better to move this to a separate question.
Let me call a linear subspace $V \subset L^2(0,1)$ to be tame if, for every linear subspace $W \subset V$, either $W$ is dense in $L^2(0,1)$, or ...
- 13.8k
9
votes
2
answers
775
views
Heat flow, decay of the Fisher information, and $\lambda$-displacement convexity
In the whole post I will work in the flat torus $\mathbb T^d=\mathbb R^d/\mathbb Z^d$ and $\rho$ will stand for any probability measure $\mathcal P(\mathbb T^d)$. This question is strongly related to ...
- 5,380
9
votes
1
answer
429
views
Is $\mathcal{S}(\mathbb{R}^n)$ a tame Fréchet space?
Hamilton's paper "The Inverse Function theorem of Nash and Moser" (1982, Bull. Amer. Math. Soc, vol. 7, n. 1, page $137$) proves that $C^{\infty}(M)$ is a tame Fréchet space when $M$ is a compact ...
- 370
9
votes
4
answers
2k
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How may I find all continuous and bounded functions g with the following property?
Find all continuous and bounded functions $g$
with :
$$\forall x \in \mathbb R, 4g(x)=g(x+1)+g(x-1)+g(x+\pi)+g(x-\pi).$$
I have posted this question here, but received no answer.
- 4,074
9
votes
2
answers
849
views
$\zeta$-function regularized determinants
In (mathematical) physics in order to compute path integrals one often makes an infinite dimensional change of variables and uses infinite Jacobian as a purely formal expression. This step is done in ...
- 21.8k
9
votes
2
answers
4k
views
Negative real order Sobolev spaces: density and representation
First, I give my motivation to ask this question. The generalised Neumann trace can be defined as
$$
{}_{H^{-1/2}(\partial\Omega)}\langle\frac{\partial u}{\partial{\mathbf{n}}},v\rangle_{H^{1/2}(\...
- 291
9
votes
5
answers
870
views
Abelianization of GL(H)
This is related to Theo's question about the abelianizations of finite dimensionsal Lie groups.
I am interested in a specific (infinite-dimensional) case of the above question. Let H be an infinite-...
- 629
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votes
2
answers
1k
views
Borsuk pairs of Banach spaces
Given $X$, $Y$ two real Banach spaces, let's say that $(X,\ Y)$
is a Borsuk pair if for any continuous mapping $T$ : {$x$ $\in$
$X$ ; $||x||\leq1$} $\rightarrow$ $Y$ s.t. $T$ is odd on {$x$
$\in$ $X$ ;...
- 4,060
9
votes
1
answer
840
views
Conformal changes of metric and geodesics
Suppose $(M,g)$ is a Riemannian manifold. Let us assume that $X$ denotes a vector field in this manifold and consider the integral curves of this vector field.
Does there exist a conformal factor $c$ ...
- 4,153
9
votes
4
answers
1k
views
Boundedness of nonlinear continuous functionals
Let $K$ be the closed unit ball of $C[0,1]$, and let $f$ in $C(K,\mathbb{\, R})$.
Is it true that there exists an infinite dimensional reflexive subspace
$E$ of $C[0,1]$ s.t. $f(K\cap E)$ is bounded ?
...
- 4,060
9
votes
1
answer
338
views
Commuting nets for commuting projections
I think this should not be too difficult, but I am not an expert. I did not get an answer on stackexchange.
Let $A$ be a $C$*-algebra and let $p,q\in A^{**}$ be two commuting projections. Then there ...
- 633
9
votes
3
answers
2k
views
Real analyticity of solution of heat equation
Consider the heat equation $\partial_t u - \Delta u = 0, u(0, x) = u_0$ on a complete (non-compact) Riemannian manifold $M$, may be even $\mathbb{R}^n$. I was wondering, what are some known sufficient ...
- 1,407
9
votes
1
answer
609
views
Long-time decay of heat kernel on compact manifolds
Let $M$ be a compact Riemannian manifold and let $V \in C^\infty(M)$. Consider the operator $\Delta + V$ and let $p_t(x, y)$ be the corresponding heat kernel. If $\Delta + V$ is a positive operator ...
- 9,853
9
votes
1
answer
1k
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A question concerning separate and joint continuity of bilinear maps
Suppose that $V$ is a locally convex topological vector space and $f:V^2 \to V$ is a bilinear map. Suppose that $C \subseteq V$ is compact and convex, $f$ maps $C^2$ into $C$ and
$f \restriction C^2$ ...
- 3,547
9
votes
1
answer
359
views
Relaxation of notion of positive definite function
A function $f:\mathbb{R}\to\mathbb{R}$ is called positive definite (in the semigroup sense) if for all $n\geq 1$ and $x_1,\ldots,x_n\in\mathbb{R}$ pairwise different the matrix $(f(x_i+x_j))_{i,j=1}^n$...
- 3,031
8
votes
1
answer
386
views
Lower bound for $\frac{\sum_{i,j}\min((f_i-f_j)^2,(g_i-g_j)^2)}{\sum_{i,j}\max((f_i-f_j)^2,(g_i-g_j)^2)}$
Let $f\in\mathbb{R}^n$ and $g\in\mathbb{R}^n$ be two orthogonal unit vectors such that $\sum_{i}{f_i}=\sum_{i}{g_i}=0$.
Question. Can we prove this?
$$\frac{\sum_{\{i,j\}}\min((f_i-f_j)^2,(g_i-...
- 519
8
votes
1
answer
341
views
characterization of commutative Banach algebras
Let $A$ be a Banach algebra with the following property:
For every two nets $ x_{\alpha}$ and $y_{\alpha}$ in $A$, $x_{\alpha}y_{\alpha}$ converges if and only if $y_{\alpha}x_{\alpha}$ converges.
...
- 356
8
votes
0
answers
6k
views
Convex hulls of compact sets
Let $A$ be a compact set in a separable Hilbert space $H$, and let $\bar A$ denote its convex hull. Is $\bar A$ compact?
- 8,512
8
votes
2
answers
471
views
Obstructions for the wedge of coordinate differentials to be harmonic
Let $(M,g)$ be a smooth $d$-dimensional Riemannian manifold, $d$ even. Are there obstructions (I guess in terms of curvature) for $g$ to have the following property:
For every $p \in M$ there exist a ...
- 6,741
8
votes
1
answer
485
views
An inequality related to Riesz–Thorin theorem, determinants and $L_p$ norm
Let $a, b, c \in \mathbb{R}^n$ , $p \in [1, +\infty)$, prove that
$$\left( \sum_{1\leq i < j <k \leq n} \left| \det\left(\begin{matrix} a_i & b_i & c_i \\
a_j & b_j & c_j \\
...
- 563
8
votes
2
answers
644
views
In which sense are Euler-Lagrange PDE's on fiber bundles quasi-linear?
In what follows, all manifolds are smooth, Hausdorff, paracompact, connected and oriented, and all maps between any two of them are assumed to be smooth. Let $\pi:E\rightarrow M$ be a fiber bundle ...
- 7,817
8
votes
3
answers
2k
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The "Spaces of Schwartz distributions are finite dimensional" challenge
The more I study Schwartz distributions and the corresponding spaces, the more the latter look "finite dimensional" to me. Of course they are not finite dimensional in the technical sense but they are ...
8
votes
2
answers
3k
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$L^p$-norm of Fourier series in terms of coefficients, $p \neq 2$
It is known that the $L^2$-norm of a Fourier series equals the $l^2$-norm of the coefficients. Are there similar results in the case of $L^p$-norm for $p\neq 2$? Can it be expressed explicitly in ...
- 176
8
votes
1
answer
268
views
Two questions about basic sequences
Suppose $(x_n)$ and $(y_n)$ are two basic sequences in a separable Banach space $X$ such that $\overline{span}\{(x_n), (y_n)\}=X$. Can we always pass to subsequences $(x_{n_k})$ and $(y_{n_k})$ such ...
- 355
8
votes
0
answers
260
views
Hyperbolic PDEs - Proof that the restriction of a locally $H^s$ solution to a spacelike hypersurface is locally in $H^s$
I have found the following claim made very clearly at least once in the published literature (see below):
Let $P$ be a linear partial differential operator defined on an open set $\Omega \subset \...
- 503
8
votes
1
answer
523
views
Are the following subsets of a Hilbert space always homeomorphic?
Let $F$ be a infinite-dimensional complex Hilbert space, with inner product $\langle\cdot\;| \;\cdot\rangle$, the norm $\|\cdot\|$, the 1-sphere $S(0,1)=\{x\in F;\;\|x\|=1\}$ and let $\mathcal{B}(F)$ ...
- 724
8
votes
0
answers
191
views
Lifting a determinant map
This is a kind of a follow-up to Question on Hessian of a function (probability question). Suppose I give you a continuous function $f:\mathbb{R}^n \to \mathbb{R}.$ Is it true that there exists a ($C^...
- 96.4k
8
votes
3
answers
858
views
What does the flow of the principal symbol of the differential operator tell us about the PDE?
Disclaimer: Let me apologize in advance for asking this slightly vague question
Let $M$ be a manifold and let $P$ be a partial differential operator acting on $C^{\infty}(M)$. Associated to $P$ there'...
- 7,789
8
votes
1
answer
1k
views
Is the p-norm of a matrix strictly log-convex?
Let $A$ be a $n\times n$-matrix. We let $\|A\|_p$ denote the norm of $A$ when considered as a linear operator on $\ell^p(\{1,2,\ldots,n\})$, that is,
$$
\|A\|_p = \sup_{x\neq 0}\frac{\|Ax\|_p}{\|x\|_p}...
- 3,497
8
votes
2
answers
488
views
If the cardinality of $B(X)$, the space of operators on $X$, is continuum, must $X$ be separable?
Does there exits any non-separable Banach space $X$ such that the size (cardinal number) of $B(X)$, bdd linear operators on $X$, is just of the continuum?
- 4,058
8
votes
1
answer
406
views
How to solve the system of PDEs defining Killing vectors
Recently I came across the following problem. Here's the setting:
Let $(M^n,g)$ be a Riemannian manifold, $\nabla$ the Levi-Civita connection, and $U$ a coordinate neighbourhood with coordinates $\{x^...
8
votes
3
answers
1k
views
Dual Banach space of $B(X,Y)$ when $X$ is finite dimensional
Denote $B(X,Y)$ the Banach space of bounded operators between Banach spaces $X$ and $Y$.
When $X$ and $Y$ are both finite dimensional, it follows from the formula
$$\|u\|_{B(X,Y)} = \sup_{\|x\|_X <...
- 9,486
8
votes
1
answer
455
views
Closure of $L(\ell^2,\ell^2)$ in $L(\ell^2,\ell^\infty)$
Let $\ell^2$, $\ell^\infty$ denote the usual sequence spaces and let $L(\ell^2,\ell^2)$ the Banach space of bounded linear operators from $\ell^2$ to $\ell^2$ as well as $L(\ell^2,\ell^\infty)$ the ...
- 261
8
votes
2
answers
1k
views
What is the smallest $C^*$-algebra containing the "standard" pseudodifferential operators?
Is $\Psi^0(\mathbb{R})$ (pseudodifferential operators with symbols obeying
$
|\partial^\alpha_x \partial^\beta_\xi a(x,\xi)| \leq C_{\alpha,\beta} (1+|\xi|)^{-|\beta|}
$
) a $C^*$-algebra?
In other ...
- 7,197