All Questions
13,927 questions
1
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Simple convergence of convex compact set implies Hausdorff convergence
I am wondering about the following :
In $\mathbb{R}^n$, suppose you are given compact convex bodies $\left\{ C_k : k \geq 1 \right\}$ and $C$, such that for every $x \in \mathbb{R}^n$ $$ \mathbb{1}_{...
- 125
1
vote
1
answer
53
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Zeros of Gram-Schmidt derived polynomials in weighted integral space
This is a problem out of Chapter 3 of Luenberger's Optimization by Vector Space Methods that I have been having trouble with. Any guidance would be appreciated.
Let $w(t)$ be a positive (weight) ...
- 136
0
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0
answers
22
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Directions of differentiability of log-concave measures with infinite-dimensional support
I recently came across the very nice review "Differentiable Measures and the Malliavin Calculus" by Bogachev (1997) which begins by discussing measures $\mu$ on locally convex spaces $X$ ...
- 661
-1
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2
answers
232
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Determining if $\|f\|_\infty \leq C\, \|f\|_{2}^{2/3} $ holds under $f(0) = f(1) = 0$, $\|f'\|_2 \leq 1$
Suppose $f \colon [0, 1] \to \mathbb{R}$ is continuously differentiable, and satisfies $f(0) = f(1) = 0$ and $\|f'\|_2 \leq 1$.
I am wondering if it there is a constant $C > 0$ such that for all ...
- 6,486
5
votes
1
answer
311
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Quotients in categories of metric spaces
There are several categories whose objects are metric (or pseudo-metric) spaces. Natural choices of morphisms are continuous, uniformly continuous, Lipschitz or short (= non-expansive or contractive) ...
- 5,596
0
votes
1
answer
121
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A simple bilinear estimate
Let $2\leq p,q <\infty$ and fix $0<\alpha<1$ such that
$\frac{1}{p}+\frac{1}{q}\leq 2-\alpha$.
Suppose that $f\in L^{p}([0,1])$ and $g\in L^{q}([0,1])$.
What is the optimal value of $t=t(\...
- 852
2
votes
0
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88
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An example of an $\mathcal{L}_\infty$ Banach space with property p-(V) and without property (V)
Here are the definitions for property $p$-$(V)$ and property $(V)$.
A Banach space $X$ has property $(V)$ if and only if every unconditionally converging operator $T$ from $X$ to any Banach space $Y$ ...
2
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1
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320
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Fourier series but different waveform
Given a nondegenerate smooth simple closed convex curve $f: [0,2\pi]\to \mathbb C \setminus \{0\}$ with winding number (around origin) $1$, and $f$ have zero mean. Let $f_n: [0,2\pi]\to \mathbb C \...
-3
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1
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76
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Minimal norm problem with linear combination of translation operator to be estimated
Follow up question from this one
Suppose $X = L^2(G)$, where $G$ is some locally compact group. Let $x, y \in G$ I for fixed $n$ I am seeking for an operator $H \in B(X)$ of the form
$$
H = H(\alpha_1,...
- 39.1k
3
votes
2
answers
392
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Monotonicity of matrix conjugation
Let $A$ and $B$ be positive-definite matrices such that $A \le B.$
By matrix monotonicity of the root, this also implies that $A^{\alpha} \le B^{\alpha}$ for $\alpha \in [0,1].$
I am now curious under ...
- 6,351
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0
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80
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Verifying the Cauchy behavior of a sequence
Let me consider the iteration $x_{n+1}=Tx_n$ and $T$ is a self-map from a non-empty subset $K$ of a smooth Banach space $X$ to itself, satisfying $W(Tx, Ty) \leq W(x, y)$, where $W(x, y)=\Vert x \Vert^...
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1
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118
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Minimal norm problem whose unknown is an operator
Generally given an Hilbert space $X$ with and a bounded linear operator $H : X \to X$ given a vector $y \in X$ we seek an $x \in X$ such that
$$
f(x) = \frac{1}{2} \left\lVert Hx - y \right\rVert_2^2
$...
- 39.1k
5
votes
1
answer
188
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On a property for normed spaces
I asked this question on Math Stackexchange, but I didn't get an answer:
https://math.stackexchange.com/questions/4881155/on-a-property-for-normed-spaces?noredirect=1#comment10410489_4881155
I came ...
- 1,848
7
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151
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Stochastic analysis on nuclear Fréchet spaces
This is a reference request question, so to make it clear what I am after, I will give a quick outline of the area I am thinking in and some questions that arise.
A lot of the time in infinite-...
- 439
1
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1
answer
180
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Shape, shift and scaling retrieval of a sampled function
Let $f(x)$ be some unknown continuous square-integrable function defined on the interval $[0,1]$.
Suppose we have $i=1,...,n$ samples $f_i$ of $f$ of the following form:
$$f_i(x)=a_i*f(x+b_i)+c_i$$
...
CommunityBot
- 1
0
votes
3
answers
753
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center of the algebra of bounded operators [closed]
Suppose that $X$ is a Banach space. How to prove that the center of the algebra $B(X)$ (the algebra of bounded operators on $X$) consists only of operators of the form $aI$, where $a$ is scalar and $I$...
- 6,367
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0
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32
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reference request: mercer expansion and kernel underlying Sobolev spaces?
Let us define the periodic Sobolev spaces, for $s > n/2$ by
$$
H_{s}([0, 1]^n) = \{f : [0, 1]^n \to \mathbb{R} :\mbox{for}~j\leq s, f^{(j)} |_{\partial[0, 1]^n} \equiv 0, ~~
\int_{[0, 1]^d} (f^{(s)...
- 380
12
votes
2
answers
5k
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Where was/is Compensated Compactness used?
This last summer, I read up on Tartar's so called Method of Compensated Compactness (or at least how it applied to scalar conservation laws). I used this theory to prove the existence of $L^{\infty}$ ...
- 4,713
3
votes
2
answers
774
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Question about closed projection
I'm wondering if the following can be true:
Let Y be a second countable space and
$\pi_2:Y \times \mathbb{R}\rightarrow\mathbb{R}$ ($\mathbb{R}$ with its usual topology and
$\pi_2$ the projection onto ...
- 1,446
0
votes
0
answers
150
views
Connectedness of deleted symmetric product
Let $X$ be a connected Hausdorff space. It is well-known that the $n$-fold symmetric product $\mathcal{F}_n(X) := \{A\subseteq X : 0<|A|\leq n\}$ is a connected space equipped with the Vietoris ...
- 10.9k
5
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1
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696
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Is any order bounded continuous linear functionals a difference of positive continuous functionals?
Let $B$ be a Banach space and $K$ a closed proper cone in $B$ such that the induced partial order makes $B$ a vector lattice. Let $K'=\{x^*\in B':\langle x^*, x\rangle\geq 0\ \forall x\in K\}$ the ...
- 4,219
1
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1
answer
132
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Variants of Dirichlet-type function as a pointwise limit of continuous functions
Problem
Suppose $f$ is a function from a complete metric space $X$ to a metric space $Y$, and suppose $Y$ has points $y_{0}$, $y_{1}$ such that the subsets $f^{-1}(y_{0})$ and $f^{-1}(y_{1})$ are both ...
- 5,409
0
votes
0
answers
55
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Strong sub-differentiability of an equivalent strictly convex norm
First, we define the notion of strong sub-differentiability(SSD) of a norm on a Banach space $X$. The norm $\Vert \cdot \Vert$ of $X$ is said to be SSD if the one-sided limit $$\lim_{t \to 0+} \frac{\...
- 85
2
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0
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414
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$$ \left(\frac{\text{Man}^{\text{fr}}}{\text{Cobordism}},\coprod,\times \right)\simeq \left((\text{Fin}^{\simeq},\coprod)^{\text{gp}},\times\right)?$$ [closed]
If we combine a theorem of Pontryagin and the Barratt-Priddy-Quillen theorem we get that both sides of
$$
\left(\frac{\mathrm{Man}^{\mathrm{fr}}}{\mathrm{Cobordism}},\coprod,\times \right)\simeq \left(...
- 705
4
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1
answer
138
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Fredholm property of linearization of Floer map
I am reading Audin and Damian's book "Morse theory and Floer homology". In Proposition 8.1.4 which reveals the transversality property of moduli space of solutions of Floer equation, the ...
- 9,580
1
vote
1
answer
67
views
Norm of differentiation operator with respect to Gaussian norm
Here is a problem from Luenberger's optimization by vector space methods. I would appreciate steps to proceed.
Let $\mathcal{P}_n\subset\mathbb{R}[x]$ be polynomials of degree at most $n\ge0$. Compute ...
- 24.2k
0
votes
1
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140
views
Approximating a sequence of tempered distributions "uniformly" by Schwartz functions
This question has been motivated by the post making sense of distributions on the diagonal.
Let $T$ be a tempered distribution on $\mathbb{R}^2$ and $\eta$ be a given mollifier on $\mathbb{R}$. For $f ...
- 3,584
2
votes
0
answers
103
views
What $1$-forms $\theta$ solve $\Delta \theta = f\theta$ for a smooth function $f$?
I have a seemingly basic question that I cannot find any literature on. Let $(M,g)$ be a smooth Riemannian manifold and let $\Delta:\Omega^1(M) \to \Omega^1(M)$ be the Laplace-De Rham operator on $1$-...
CommunityBot
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2
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1
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179
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Factorization systems for vector bundles
Are there any well-known factorization systems for the category of vector bundles defined over topological spaces?
- 30.3k
6
votes
1
answer
149
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Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ is surjective?
Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ is surjective ?
- 18.6k
2
votes
0
answers
35
views
Continuity of Kernel Mean Embeddings
Given some kernel $k: X \times X \to \mathbb{R}$ with RKHS $H_k$ we say that $k$ is characteristic on the space of signed Radon measures over $X$, denoted by $\mathcal{M}(X)$, if the kernel mean ...
- 161
1
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1
answer
182
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Inequalities involving entropy: quantum discord and mutual information
My question is inspired by the following paper of Olivier and Żurek but for this question to be self-contained I will recall all the necessary definitions: for a quantum state $\rho$ we define the ...
- 9,330
1
vote
0
answers
81
views
"Star" of a CW-complex
Suppose we have a CW-complex $X$ with a 0-cell $e^0$. Is the union of all the cells (of higher dimensions) for which $e^0$ is a boundary point open in $X$?
I don't know if it has a name, but a similar ...
- 3,065
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1
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78
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Let K be a compact set in a surface, U component of S-K, K'=S-U. K has finitely many components. Does every component of K' contains a component of K? [closed]
Let $S$ be a compact connected surface. Let $K$ be a compact subset of $S$ and suppose that $K$ has a finite number of connected components.
Let $U$ be a connected component of $S \setminus K$ and ...
- 3,599
3
votes
1
answer
177
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Is there a metric separable space with the following properties...?
Let $\omega_1<\mathfrak{q}_0$ where $\mathfrak{q}_0:=\min\{|Y|:Y\subseteq \mathbb{R}$, $Y$ is not a $Q$-space$\}$.
Is there a metric separable space $X$ with the following properties:
$|X|\geq\...
- 11.4k
4
votes
3
answers
482
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Does the uniform boundedness principle holds for multilinear maps as well?
This question has been motivated by weak* completeness of distributions.
According to the answer in the above post, any barrelled locally convex topological vector space $E$ satisfies the uniform ...
- 3,584
2
votes
1
answer
128
views
On the existence of a complicated fractal-like set of finite perimeter
Let $f\in BV(\Bbb R^n)$ be an integer-valued function that maps into $\{0, 1\}$ and is identically $0$ outside some bounded set in $\Bbb R^n$. In particular, $f$ determines a bounded Caccioppoli set $...
- 28k
2
votes
2
answers
343
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Does locally compact plus pseudocompact imply paracompact?
This one is probably simple, but I don't see it yet.
Is a locally compact, pseudocompact Hausdorff space necessarily paracompact?
- 4,713
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votes
0
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48
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Computation of Brown measure of the shift operator on $\ell^2(\mathbb N)$?
This looks an extremely simple question - I am just trying to give an example of Brown measure, https://en.wikipedia.org/wiki/Brown_measure, so I try to compute it for the left/right-shift operator on ...
- 1,755
2
votes
1
answer
232
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Banach spaces locally having a basis
The $\mathcal{L}_p$-spaces ($1\leq p \leq \infty$) are Banach spaces $X$ such that there exists a constant $\lambda$ so that every finite dimensional subspace $E$ of $X$ is contained in another ...
- 2,605
3
votes
1
answer
3k
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Approximate point spectrum
I have a question concerning the relation between the approximate point spectrum and the spectrum of an operator.
Let $T$ be a bounded linear operator of a complex Hilbert space $H$. The approximate ...
CommunityBot
- 1
1
vote
0
answers
109
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PDE coupled with the pronic numbers (related to triangular numbers)
I am studying the linear PDE:
$$ t^2\frac{\partial^3}{\partial t^3}\sum_{n=1}^\infty \Psi_n(t,s)=s^2\frac{\partial}{\partial s}\sum_{n=1}^\infty \Psi_n(t,s)+\sum_{n=2}^\infty b(n)\frac{\partial}{\...
- 383
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5
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830
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How can one characterise compactness-by-experiment?
There are a myriad different variations on the theme of "compactness", and some of them have even made it on to Wikipedia. I'm interested in finding out more about types of compactness that ...
- 1,446
3
votes
0
answers
94
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Harmonic heat flow, formal and rigorous
Let $ (M,g) $ be a smooth Riemann manifold without boundary, $ S^{n-1} $ is an $ n $-dimensional sphere, and $ T>0 $. Consider a weak solution $ u:M\times[0,T]\to S^{n+1} $ of
$$
\partial_tu-\Delta ...
- 519
2
votes
2
answers
155
views
"Completeness" for weak convergence of unbounded closed operators on a separable Hilbert space $H$
Let $H$ be a separable Hilbert space with the inner product $\langle, \rangle$ and $\{ T_n \}$ be a sequence of unbounded closed linear operators with a common dense domain $D \subset H$ such that $...
- 24.2k
7
votes
1
answer
361
views
What is the analogue of the Jacobi theta function in the Weyl representation?
It is known (see for example the associated Wikipedia entry) that the Jacobi theta function
$$\vartheta(z; \tau) = \sum_{n\in\mathbb{Z}} \exp(\pi in^2\tau + 2\pi inz)$$
arises from a certain ...
- 6,367
1
vote
1
answer
110
views
Existence of a Hölder homeomorphism satisfying prescribed norm constraints
Let $\Omega$ be a convex body$^{\boldsymbol{1}}$ in $\mathbb{R}^n$ where $n$ is a positive integer. Fix a positive integer $k$ and some $0<\alpha\leq 1$. Let $k_1> k_2>0$. Does there ...
- 60.6k
2
votes
2
answers
235
views
$L^p$ domination of mixed partial derivatives of the 3rd order by the unmixed ones?
Is it true that for each real $p>1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^3$ one has
$$\|D_1D_2D_3u\|_p\le C_p(\|D_1^3u\|_p+\|...
- 128k
1
vote
1
answer
133
views
A question about the maximal function
Let $n>4$, $f\in C^{\infty}(\mathbb{R}^{n})$ and 0 denote the origin of $\mathbb{R}^{n}$. We define a weighted maximal function by $$Mf(x)=\sup_{0<r<1}r^{4-n}\int_{B_{r}(x)}|f|$$ which is ...
- 28k
20
votes
2
answers
1k
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P-adic C* algebras
I understand that there is a definition of p-adic Banach algebras and that a significant amount of functional analysis can be developed in the non-archimedean setting. Is there a p-adic version of C*-...
- 1,485