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Is any submetrizable linear topology linearly submetrizable?

Let $E$ be a vector space. A topology $\tau$ on $E$ is called (linearly) submetrizable if there is a (linear) metrizable topology $\pi$ on $E$ which is weaker than $\tau$, i.e. $\pi\subset\tau$. Is ...
erz's user avatar
  • 5,529
1 vote
0 answers
75 views

Banach lattice hull of a Banach space

I am interested in defining some Banach lattice properties for subsets of arbitrary Banach spaces. So it would be useful to have the notion of Banach lattice hull $E(X)$ of a Banach space $X$. I ...
M.González's user avatar
  • 4,461
3 votes
1 answer
187 views

Is this property preserved under weak$^*$ convergence?

Let $1 \le p < n$ and let $p^*$ be the Sobolev conjugate of $p$, i.e. $p^* = np/(n - p)$. Let $(\Omega_m)$ be an increasing sequence of bounded, convex and open sets such that $$ \lim_{m \to \infty}...
Cauchy's Sequence's user avatar
2 votes
1 answer
89 views

Upper bound on the Levy-Prokhorov distance between the distributions of continuous Gaussian processes in terms of their covariances

Denote by $d$ the supremum metric on the space $C[0,T]$ of continuous real-valued functions on $[0,T]$: $$ d(f,g) = \sup_{t \in [0,T]} |f(t)-g(t)|. $$ Let $\rho$ be the Levy-Prokhorov metric on the ...
ssss nnnn's user avatar
  • 177
0 votes
1 answer
96 views

Existence of a complemented basic sequence

Let $X$ be an infinite-dimensional Banach space (complex or real). A subspace of $X$ means a closed linear submanifold. If $S$ is a non-empty subset of $X$, then $[S]$ denotes the closed linear span ...
Janko Bracic's user avatar
3 votes
1 answer
176 views

Are measurable maps with countably separated image in a Banach space always strongly measurable?

Let $(E,\|.\|)$ be a (not necessarily separable) Banach space and $\Sigma_E$ the Borel $\sigma$-algebra (w.r.t. the norm topology). Let $(\Omega,\Sigma_\Omega)$ be a measurable space (which we can ...
Packo's user avatar
  • 285
1 vote
0 answers
87 views

Convergence and sequential compactness for nonlinear operators

I have a family of operators $T_n\colon X \to Y$ where $X,Y$ are Hilbert spaces. These operators are nonlinear. What kind of notions of convergence does one have for such operators? I'm specifically ...
C_Al's user avatar
  • 251
3 votes
0 answers
90 views

Sobolev embedding on a compact manifold without boundary

I am reading M. E. Taylor, "Partial Differential Equations III", Second Edition, Springer-Verlag, New York, (1996). In chapter 13, section 2, in Prop. 2.3 and Prop. 2.4, one finds the ...
Azam's user avatar
  • 311
3 votes
1 answer
104 views

From Wightman to HK axioms for "non-neutral (charged?)" fields

Wightman axioms deal with operator-valued distributions (Wightman fields) whose values are unbounded operators in general. On the other hand, the Haag-Kastler axioms deal with net of observables, ...
Isaac's user avatar
  • 3,477
2 votes
1 answer
474 views

Polynomial $f(x)$ has positive coefficients and only real roots. How many polynomials formed from terms of $f(x)$ also have only real roots?

Let $$f(x)=a_n \ x^n+a_{n-1} \ x^{n-1}+\cdots+a_1 \ x+a_0$$ be a $n$-th degree polynomial with positive coefficients such that all of its roots are real. Choose any number terms from this expression ($...
Balaji Mallikarjun S's user avatar
0 votes
0 answers
49 views

Kadec-Klee property of an equivalent norm on a Hilbert space

Let us consider the space $\ell_2$ with the Hilbert norm $\Vert \cdot \Vert$ and consider the following eqivalent norm: $$ \Vert (r,x) \Vert_A^2 = \Vert (r, Tx)\Vert^2 + \max \{ \Vert x \Vert, \vert r ...
PPB's user avatar
  • 85
2 votes
0 answers
103 views

A question from a proof of an inequality in Sobolev space $W^{1,1}$

I try to understand the proof the lemma given at page 54 in Ladyzhenskaya et al (1968) - Linear and Quasilinear Elliptic Equations. Here it is a screenshot: Here is what I did: $$-u(x)=u(y)-u(x)=\...
Bogdan's user avatar
  • 1,759
3 votes
1 answer
327 views

Derivative norm estimates

Assume $\Phi$ is some diffeomorphism of a certain manifold. Let $\Phi^{-1}$ denote the inverse map and let $(D\Phi)^{-1}$ denote the matrix inverse of $D\Phi$. QUESTION. Does this norm estimate hold? ...
T. Amdeberhan's user avatar
2 votes
1 answer
127 views

Strong Ditkin sets in the Fourier algebra

What is the definition of a Ditkin set (resp. a strong Ditkin set) for the Fourier algebra $A(G)$ of a locally compact (not necessarily abelian) group $G$? More specifically, let $E$ be a closed ...
Aristides's user avatar
4 votes
0 answers
140 views

Condition under a function is uniquely identifiable by the supremum values

Let $f(x),g(x)$ be two real-valued functions on $\mathbb{R}$ and $h(x,y)$ be a real-valued function on the plane. We can assume continuity (maybe piecewise differentiability also) of these functions. ...
mukhujje's user avatar
  • 271
1 vote
0 answers
67 views

regularity convolution of a $L^2$ function with $W^{1,1}$ function [closed]

Let $u\in L^2(\mathbb R)$ and $w \in W^{1,1}(\mathbb R)$, we consider the convolution $$u*v$$ Is it true that $w*u \in W^{1,2}(\mathbb R)$? What regularity can we put on $w$ for this to be true?
user3177306's user avatar
6 votes
0 answers
162 views

Dual space of local Sobolev space on a manifold

$\newcommand{\comp}{\mathrm{comp}}$As part of my master's thesis, I am currently learning about Sobolev spaces on manifolds. From my research online, I found out, that there are a lot of ways to ...
Fabian Patzwaldt's user avatar
1 vote
1 answer
65 views

Reference dual Dirichlet space $D^1$

Let $\mathbb{D} = \{ z \in \mathbb{C} : |z| < 1 \}$ be the unit disk. The Bergman space $A^1 = A^1(\mathbb{D})$ is the Banach space of holomorphic functions on $\mathbb{D}$ such that $$ \|f\|_{A^1} ...
Scottish Questions's user avatar
0 votes
0 answers
84 views

Question on approximation of norms

Suppose that $E\in Int[L_{p},L_{q}]$ for some $1<p<q<\infty$ and $E$ is $w$-concave with $1<w<\infty$. It is well-known that for each $r\geq w$, we have $E=L_{r}\odot F_{r}$ for some ...
Sijie Luo's user avatar
3 votes
0 answers
53 views

Bounds on Besov norms for mollification of a bounded Lipschitz function

Let $\Omega$ be a bounded, non-empty, regular open domain in $\mathbb{R}^d$. Let $1\le p,q\le \infty$ and $s>0$. Let $\mathcal{B}_{p,q}^s(\Omega)$ be the Besov space on $\Omega$ corresponding to ...
LittleQuestionBoy's user avatar
4 votes
3 answers
308 views

Intriguing simple question about Sobolev space $W^{1,p}(\Omega)$

Let $w_1,w_2\in W^{1,p}(\Omega)$ be two functions with $w_1,w_2>0$ and $\dfrac{w_2}{w_1},\dfrac{w_1}{w_2}\in L^{\infty}(\Omega)$, where $\Omega\subset\mathbb{R}^N$ is a bounded domain (i.e. open ...
Bogdan's user avatar
  • 1,759
12 votes
1 answer
402 views

Boundedness of sequences and cardinality

Let $X$ be a set of sequences of real numbers that converge to zero with the property that for any unbounded sequence of real numbers $(y_n)$, there is a sequence $(x_n)$ in $X$ for which the ...
Chris Stuart's user avatar
7 votes
1 answer
243 views

Isoperimetric inequality, but $L_p$ norm

I would like to consider the isoperimetric problem of $L_p$ norm: Given a region in $\mathbb R^2$ such that the boundary is a curve $C(x,y)$, where $\int_{C}(|\mathrm dx|^p+|\mathrm dy|^p)^{1/p}$ is a ...
JetfiRex's user avatar
  • 843
4 votes
1 answer
252 views

Show that $\Lambda_\varphi(x_n)\to \Lambda_\varphi(x)$ for an nsf weight $\varphi$ on a von Neumann algebra

Let $\varphi$ be an nsf weight on a von Neumann algebra $M$. Fix a square-integrable element $x\in \mathscr{N}_\varphi$. Put $$x_n := \sqrt{\frac{n}{\pi}}\int_{-\infty}^{+\infty} \exp(-nt^2) \sigma_t^\...
Andromeda's user avatar
  • 175
1 vote
1 answer
130 views

Existence of solutions to a series of integral equations

I am trying to solve the following integral equation analytically: $$ \sum_{n \geq 1} \left( \int_0^te^{-n^2(t-s)} f_n(s) \, ds \right) = g(t), \quad t \in [0, T], $$ where $(f_n(t))_n$ is the unknown ...
Gustave's user avatar
  • 617
7 votes
1 answer
281 views

Norm in the minimal tensor product of C*-algebras

Let $A$ and $B$ be two $C^*$-algebras, and let $A \otimes B$ denote their minimal tensor product. Given positive, linear functionals $\varphi$ on $A$ and $\psi$ on $B$, we obtain a positive, linear ...
Hannes Thiel's user avatar
  • 3,497
0 votes
0 answers
119 views

Boundedness of 2 times the unit ball

Suppose that $X$ is a topological vector space where the topology is given by a metric $d$ on $X$. Assuming that the unit ball $$ B(0, 1) := \{x \in X : d(0, x) < 1\} \neq X, $$ is it necessarily ...
Chandan Biswas's user avatar
1 vote
2 answers
220 views

A question on unit norm elements of $\ell^2 \setminus \bigcup_{0<p<2 }\ell^p$

Let $A\subset \ell^2$ consist of all $x\in \ell^2$ with $|x|_2=1$ which does not belong to any $\ell^p$ for all $0<p<2$. Note that $A$ is non-empty with a Baire category argument. I ...
Ali Taghavi's user avatar
4 votes
1 answer
196 views

(Lattice approximation) Does UV stability lead to continuum limit of a subsequence?

In the context of lattice approximation, the term "UV stability" seems to be used frequently. To me, it seems like Uniform boundedness of the partition function in the limit where lattice ...
Isaac's user avatar
  • 3,477
0 votes
0 answers
36 views

Derivate involving Bessel function of second type

Let. $$f := (x, y) \mapsto \text{BesselK}(1, c \cdot (a - b \cdot (x + y))) \cdot \exp(c \cdot b \cdot (y - x))$$ Is there a close formula for this $$\frac{\partial^{m+n}}{\partial y^m \partial x^n} f(...
Ryo Ken's user avatar
  • 109
3 votes
1 answer
376 views

A more general product rule for weak derivatives?

Consider that $u_1,u_2:\Omega\to (0,\infty)$ where $\Omega\subset\mathbb{R}^N$ is an open set. We know that $u_1,u_2\in W^{1,p}(\Omega)$ for some $p>1$ and $\dfrac{u_1}{u_2},\ \dfrac{u_2}{u_1}\in L^...
Bogdan's user avatar
  • 1,759
1 vote
0 answers
51 views

Compact embeddings RKHSs into Sobolev Spaces

Let $\mathcal{H}$ be an RKHS over an open domain $\Omega \subseteq \mathbb{R}^d$. Are there conditions under which $\mathcal{H}$ can be compactly embedded in a Sobolev space $W^{s,p}(\Omega)$ for ...
Sam_the_Sampler's user avatar
1 vote
0 answers
73 views

Convexity and subdifferential monotonicity

Do you know any reference where I can find some results in this sense: Consider $W:K\to [0,\infty)$ is a functional defined on a convex cone $K\subset X$, where $X$ is a Banach space. Then the ...
Bogdan's user avatar
  • 1,759
2 votes
0 answers
86 views

Besov spaces containing piecewise linear functions

Let $\Omega$ be a bounded, non-empty, regular open domain in $\mathbb{R}^d$. Let $1\le p,q\le \infty$ and $s>0$. Let $\mathcal{B}_{p,q}^s(\Omega)$ be the Besov space on $\Omega$ corresponding to ...
LittleQuestionBoy's user avatar
2 votes
0 answers
57 views

Mappings that preserve local or global minimum

In the most general form, I'm interested in any non-trivial results of the following question. Consider metric space $X$ and $Y$, denote all real valued functions on $X$ and $Y$ as $\mathbb{R}^{X}$ ...
patchouli's user avatar
  • 275
2 votes
1 answer
244 views

Characterization of normed spaces based on violation of parallelogram law

For a normed linear space $(X, \|\cdot\|)$, the Jordan-von Neumann theorem specifies when exactly the norm is induced via an inner product, namely when the parallelogram law is satisfied. I would like ...
Hikaru's user avatar
  • 213
1 vote
0 answers
34 views

Discrepancy between probability measures, tested against bounded functions of bounded variance

When studying some concentration inequalities, it became relevant to consider the following discrepancy between two probability measures $\pi$ and $\nu$ (treating $\sigma \in \left( 0, \frac{1}{2} \...
πr8's user avatar
  • 801
6 votes
3 answers
282 views

Extreme points of the dual unit ball of a Banach algebra

Let $A$ be a unital Banach algebra. Let $f\in A^*$, $\Vert f\Vert=1$ satisfy that there exists a maximal left ideal $L\subset A$ such that $L\subseteq\ker{f}$. Question: Is $f$ an extreme point of ...
Onur Oktay's user avatar
  • 2,605
9 votes
2 answers
418 views

Reference request: Parabolic Equations

I am a PhD student working mainly on Elliptic Equations. With the other PhDs of my department, we organised a reading group, meaning that we agreed on a book we were all interested in, we meet weekly ...
Falcon's user avatar
  • 452
5 votes
1 answer
183 views

Question about modular group (Modular theory in operator algebras, section 2.14)

Consider the following fragment from Stratila's book "Modular theory in operator algebras", section 2.14, p20: I'm trying to understand the claim $(3)$ (see the red box). The main strategy ...
Andromeda's user avatar
  • 175
3 votes
0 answers
165 views

$S^{n}(V)$ is "approximately" $V$ when $n$ goes to infinity (in the setting of normed space)

Let $B$ be a (separable) Banach space. $(v_{i})_{i}, i\in\mathbb{N}$ being a family of linear independent vectors. $V$ being the span of $v_{i}$. I try to prove that $V$ is dense in $B$. I define a ...
Ken.Wong's user avatar
  • 523
3 votes
1 answer
176 views

Question about Lebesgue Bochner spaces

Let $T>0$ and $\Omega\subset\mathbb{R}^N$ be a bounded domain. Also $p\in (1,\infty)$ is any number. I know that $u\in L^{p}((0,T);L^p(\Omega))$ and $\nabla u\in L^{p}((0,T);L^p(\Omega))^N$. How ...
Bogdan's user avatar
  • 1,759
3 votes
2 answers
137 views

Non-complete space verifying uniform boundedness

Recently, I have seen the so-called uniform boundedness theorem, which says: Let $(X,∥⋅∥)$ be a Banach space and $(Y,∥⋅∥)$ be a normed linear space. Let $A⊂B(X,Y)$ be a pointwise bounded family of ...
Tomas smith Smith's user avatar
2 votes
0 answers
92 views

Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries

For $n \geq3$, let $(M,g)$ be smooth $n$-dimensional, compact, Riemannian manifold with a smooth boundary. Then there exists some constant $A=A(M,g)>0$ such that, for all $u \in H^1(M)$ \begin{...
Arghya kundu's user avatar
2 votes
1 answer
117 views

Special density on $L^2$

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain, and $u\in L^2(\Omega)$ with $0\leq u(x)\leq 1$ a.e. on $\Omega$. It is well known that $C^{\infty}_c(\Omega)$ is dense in $L^2(\Omega)$. Because $C^...
Bogdan's user avatar
  • 1,759
11 votes
0 answers
342 views

The diagonal operators and unconditionality

The following is well-known: Theorem: Let $X$ be a Banach space with an unconditional basis $(e_n)_n$. Then the space of the diagonal operators with respect the basis $(e_n)_n$ endowed with the ...
S Argyros's user avatar
  • 986
6 votes
0 answers
159 views

Identification of Fock space and the $L^2$ space of tempered distributions

Let $\mathcal{S}'(\mathbb{R}^d)$ be the set of tempered distributions over $\mathbb{R}^d$ and $d\phi_C$ a Gaussian measure over $\mathcal{S}'(\mathbb{R}^d)$ with covariance operator $C$. Consider the ...
CBBAM's user avatar
  • 721
3 votes
0 answers
58 views

Infinitesimal generators of random evolutions

Consider two state spaces $X$ and $Y$ and infinitesimal generators of Markov processes $(A_y)_{y\in Y}$ and $B$, on $X$ and $Y$ respectively. We assume that $A_y$ share the same domain $D(A)$, and ...
Gabriel's user avatar
  • 31
4 votes
1 answer
158 views

Is the image of a complemented subspace complemented?

This question has been crossposted from mathstackexchange: Let $X, Y$ be two Banach spaces and $T:X\to Y$ a continuous surjection. Assume $Z$ is a complemented subspace of $X$ and that $T(Z)$ is ...
Pelota's user avatar
  • 655
0 votes
0 answers
59 views

Decomposition of a contractive representation into an orthogonal sum for the $n$-dimensional case. Has this been done yet?

I know that it has been done for the two-dimensional case. Marek Kosiek showed it in his work "Decomposition of operator representations of the algebra $R(K_1 \times K_2)$" and "...
S-F's user avatar
  • 63

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