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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
0 votes
0 answers
96 views

Sufficient condition for weak convergence in Banach spaces

The question is quite elementary but nonetheless no proof or counter example comes to mind immediately. Suppose that $X$ is a Banach space and $\{x_n\}$ is a sequence in $X$ such that $(x_n,y)$ ...
an_ordinary_mathematician'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
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
6 votes
1 answer
248 views

The sharpest regularity result of elliptic PDEs: conditions on the variable coefficients

Let $\Omega \subset \mathbb{R}^n$ be open and bounded with a sufficiently smooth boundary. Let $L$ be a second order differential operator with variable coefficients, given by $$Lu = \partial_i(a^{ij}...
MathsGoose's user avatar
2 votes
0 answers
47 views

Norm density of evaluation functionals in the space of weak$^*$ continuous multilinear functionals on products of dual Banach spaces

Let $K$ be a compact (metrizable) space and let $C(K)$ be the Banach space of continuous real-valued functions on $K$, equipped with the supremum norm. It is then well known that the dual space $C(K)^*...
kiliroy's user avatar
  • 56
5 votes
2 answers
517 views

Functions whose product with every $L^1$ function is $L^1$

Let $\mu$ be a probability measure and $f$ a measurable function whose product with any integrable function is integrable: $$ \int|g|\,{\rm{d}}\mu<\infty\implies \int|fg|\,{\rm{d}}\mu<\infty. $$ ...
KhashF's user avatar
  • 3,599
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
5 votes
2 answers
432 views

Does closedness of the image of unit sphere imply the closed range of the operator

Let $X$ and $Y$ be Banach spaces and let $T:X\to Y$ be a bounded linear operator such that $T(S_X)$ is closed in $Y$. Does it imply that $T(X)$ is closed? Any hint is appreciated.
Anupam's user avatar
  • 585
0 votes
0 answers
46 views

What's the problem in using spanning Bessel sequences that are not frames to decompose vectors?

This is related to a question I recently asked on math.SE. Consider a subset $G\equiv \{g_k\}_{k\in\mathbb{N} }\subseteq\mathcal H$ in a separable Hilbert space $\mathcal H$, and suppose $G$ spans the ...
glS's user avatar
  • 342
0 votes
0 answers
100 views

Construct a bi-Lipschitz mapping that maps a cube to a ball which has the same center with the cube

A mapping $f: \mathbb{R}^n\to \mathbb{R}^n$ is said to be $K$-bi-Lipschitz, $K>1$, if \begin{equation*} \dfrac{1}{K}\leqslant \dfrac{|f(x)-f(y)|}{|x-y|}\leqslant K, \end{equation*} for any $x,y\in \...
Javier's user avatar
  • 69
3 votes
0 answers
108 views

A question on essentially self-adjoint differential operators of the type $\Delta=D^{\ast}D$

Let $(M,g)$ be a smooth (connected, complete, oriented) Riemannian manifold and let $D:C^{\infty}(M)\to C^{\infty}(M)$ be a linear partial differential operator, which I view as an operator in $L^{2}(...
B.Hueber's user avatar
  • 1,171
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
1 vote
0 answers
92 views

Period of the modulus of a complex exponential sum

Consider a sum of exponentials function of the integer $x$: $f(x)=A(x)+B(x)$, where $A(x)=\sum_{i=1}^n c_i \theta_i^x$ with $\theta_i$ roots of unity, and $B(x)=\sum_{j=1}^m d_j\lambda_j^x$ with $|...
Michele's user avatar
  • 333
3 votes
1 answer
297 views

Approximation of smooth real functions with an integer integral by a sum of $\delta$-functions with coefficients $\pm 1$

This question arose from a quantum physics research (my article Entropy 2022, 24(2), 261, where I considered approximation of a smooth charge density (built from a wave function) by a collection of ...
akhmeteli's user avatar
  • 143
2 votes
0 answers
120 views

On mollifiers acting between $L^2$ and Sobolev spaces

(I'm reposting here this question from MSE as it didn't receive any answer for two weeks.) Consider a sequence of finite lattices in $\mathbb{R}^n$ defined by $$L_k= [-k,k]^n \cap 2^{-k}\cdot \mathbb{...
S.Z.'s user avatar
  • 505
2 votes
1 answer
79 views

There is some initial data such that the decay of the semigroup in it is faster than $t^{-n/2}$?

Lee and Ni show in their work Link Here that the heat semigroup $e^{t \Delta}u_0$ has decay as $t^{-\min \{a, n\} /2}$, $t \to \infty$ if $u_0 = C(1+|x|^2)^{a/2}$ if $a \neq n$. I'm trying to ...
Ilovemath's user avatar
  • 677
15 votes
2 answers
888 views

Hodge decomposition of smooth n-forms: is it an isomorphism of topological vector spaces?

Fix a compact Riemannian manifold $M$ (leaving the metric implicit). What I'd like to know is if the corresponding Hodge decomposition of smooth $n$-forms $$ \Omega^n(M) \simeq \mathcal{H}^n(M)\oplus ...
David Roberts's user avatar
  • 35.5k
3 votes
0 answers
117 views

Which sigma-ideals in a sigma-algebra are contained in an ideal of null sets?

Let $X$ be a Polish space and $\mathcal{B}(X)$ be the $\sigma$-algebra of Borel subsets of $X$. Given a Borel probability measure $\mu$ on $X$, we write $\mathcal{N}(\mu) := \{ B \in \mathcal{B}(X) : \...
Stefan Schrott's user avatar
2 votes
1 answer
142 views

Bounded differentiation operator on compact intervals with $L^2$ norm

It is known that the differentiation operator $D$ is not bounded on $C^1([0,1])$ with $L^2$ norm (counterexample: $f(x)=x^n$). Now I am wondering whether there is an infinitely dimensional subspace ...
graham's user avatar
  • 153
4 votes
1 answer
54 views

Krein-Rutman for integral transforms: proof of convergence to leading eigenvector

Disclaimer: This is a question in functional analysis, on which I don't have much background. It arose from me trying to prove on my own a folklore result in probability theory. Consider an integral ...
Plemath's user avatar
  • 312
0 votes
0 answers
112 views

Vector field connecting two points

I'm now working on somehow an inverse problem of an ODE: Suppose we have a ODE on $\mathbb{R}^{n}$: $\dot{x} = f(x)$, denote the solution to the ODE starting at $a$ as $x_{f,a}$(t). Now there is a ...
Sqr's user avatar
  • 1
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
2 votes
1 answer
75 views

How to show $\lVert\Delta u_n- \Delta u\rVert_{L^2(0,T; \,H^2(\Omega))} \to 0$ ? $(\Omega \subset \mathbb{R}^2)$

Let $u_n, \nabla u_n, \Delta u_n, \nabla \Delta u_n, \Delta^2 u_n$ be uniformly bounded in $L^2((0,T) \times \Omega)$ where $\Omega \subset \mathbb{R}^2, u=\Delta u =0$ on $\partial \Omega$. Assume ...
Arghya kundu's user avatar
0 votes
0 answers
118 views

Find the maximum of an expression under the logconcave assumption

Let $F(v)$ be a cdf over $\left[0,v_{max}\right]$, $1-F(v)$ is logconcave. The corresponding density function is $f(v)$. Let $p^m$ solve $1-F(v)-f(v)v=0$ (it is a FOC of a profit maximization problem)....
Ningjingzhiyuan's user avatar
2 votes
1 answer
49 views

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
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
7 votes
2 answers
351 views

Can the Banach algebra structure on $B(E)$ be (almost) retrieved from its Banach space structure?

This is basically just out of curiosity. Also, since my research area is in von Neumann algebras and my knowledge of general Banach algebras as well as general Banach spaces is somewhat limited, I ...
David Gao's user avatar
  • 2,830
1 vote
2 answers
209 views

Approximate simple function $f$ by a sequence of continuous functions on $\mathbb{R}^d$ such that $\|f_n\|_\infty\leq \|f\|_\infty$

Let $f=\sum_{i=1}^n c_i 1_{\Delta_i}$ be a simple function on $\mathbb{R}^d$, where $c_i\in\mathbb{C}$. Then we can find sequnces of continuous functions $\{f_k^{(i)}\}$ for each $i=1,\ldots,n$ such ...
mathlover's user avatar
1 vote
2 answers
164 views

Existence of directional heat equation without uniform ellipticity

I am asking for references, or for a proof idea on how to show that weak solutions of the following problem exist: search $u$ on a bounded domain $\Omega\times (0,T]$, where $\Omega\subset\mathbb{R}^d$...
l'étudiant's user avatar
2 votes
1 answer
99 views

A question on Bloch functions

Let $\mathcal{B}(\Delta)$ be the space of Bloch functions in the unit disk $\Delta$. For any $f\in \mathcal{B}(\Delta)$, we define the Bloch norm by $$ \|f\|_{\mathcal{B}}=\sup_{|z|<1}|f'(z)|(1-|z|^...
yaoxiao's user avatar
  • 1,706
2 votes
0 answers
83 views

3/2 Sobolev Norm on the boundary of a bounded open subset of $\Bbb R^n$

Let $\Omega\subset\mathbb{R}^{n}$ be a open bounded set and $\partial\Omega$ be the boundary of $\Omega$. Following the reference text by Alois Kufner, Oldřich John and Svatopluk Fučík, Function ...
Himanshu Garg's user avatar
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
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
2 votes
1 answer
149 views

Show that $\|P(f\circ\varphi_{\lambda})-\widetilde{f}(\lambda)\|_p=\|P(f\circ\varphi_{\lambda}-\overline{P(\overline{f}\circ\varphi_{\lambda}}))\|_p.$

Let $\Omega = \mathbb B_n,$ the unit ball in $\mathbb C^n$ and $L^2_a(\Omega)$ be the Bergman space endowed with the normalized volume measure on $\Omega.$ Let $k_{\lambda}$ be the associated Bergman ...
Anacardium's user avatar
1 vote
0 answers
55 views

Characterizing one-sided M-projections on real C*-algebras

Let $A$ be a real C*-algebra, and let $P: A \to A$ be a bounded linear projection. We say that $P$ is a left M-projection if the map $$ v_P: A \to C_2(A), \quad x \mapsto \begin{pmatrix} P(x) \\ x - P(...
Neal B's user avatar
  • 11
-1 votes
1 answer
86 views

how take weak derivative of norms in hilbert spaces?

Let the following be hilbert spaces with dens inclusions $V ↪H=H^* ↪V^*$. Where $H^*$ and $V^*$ are the duals. $H$ has the product $(*,*)$ and $V×V^*$ has the product $⟨*,*⟩$. Let $u∈L^2 ([0,T];V); ...
Alucard-o Ming's user avatar
5 votes
2 answers
258 views

Boundary value of Sobolev space

Let $D$ be a regular domain in $\mathbb R^2$. Suppose that $u \in H_0^1(D) \cap C(D)$. Does this imply $u \in C(\overline D)$ and $u|_{\partial D} = 0$?
Focus's user avatar
  • 177
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
0 votes
0 answers
80 views

Relationship between two minimization problems

Let $1 \le p < n$ and $p^* = np/(n - p)$. Let $B \subset \mathbb{R}^n$ be a closed ball and let $\Omega \subset \mathbb{R}^n$ be an open set containing $B$. We denote by $W^{1, p}_{B}(\Omega)$ the ...
Cauchy's Sequence's user avatar
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
4 votes
1 answer
219 views

Poincaré inequality on compact manifolds without boundary

The question arises from a paper The heat flow for the full bosonic string, Ann. Global Anal. Geom. 50 (2016). In line 4, Page 362, the author claimed the following inequality which looks similar to ...
user450201's user avatar
3 votes
1 answer
157 views

How can discrete Fourier transform approximation prove the completeness of complex exponentials in $L^2(T)$?

I have a question about the completeness of complex exponentials in function spaces. For the discrete set $ S = \{1, 2, \ldots, n\} $, it is clear and intuitive that $ e^{2\pi ikx/n} $ for $ k = 0, 1, ...
Zhang Yuhan'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
6 votes
1 answer
257 views

Example/Existence of Positive Linear Functional which is NOT Hermitian

We know that if $\mathcal{A}$ is a unital $C^*$-algebra and if $f:\mathcal{A}\to\mathbb{C}$ is a positive linear functional then it is Hermitian. It simply follows from the fact that in $\mathcal{A}$ ...
UtsabrajSarkar's user avatar
4 votes
3 answers
509 views

Fourier transform in $L^1$?

Let $f \in L^1 \cap L^2$. Are there any natural conditions on $f$ that ensure that the Fourier transform $\hat f$ is in $L^1?$ I don't want to have anything as restrictive as Schwartz. I am rather ...
António Borges Santos's user avatar
7 votes
1 answer
253 views

Does a Banach algebra version of "the sum of a closed subspace and a finite dimensional subspace is always closed" exist?

In the setting of Banach spaces, it is well know that if $M$ is a closed subspace of a Banach space $X$ and $F$ is a finite dimensional subspace of $X$, then $M+F$ is closed. Does a Banach algebra ...
Qingping Zeng's user avatar
4 votes
1 answer
255 views

Asymptotic behavior and of an integral on a d-dimensional torus

I am trying to evaluate the asymptotic behavior of the following integral as $t \to \infty$: $$ I(t; \mathbf{v}) = \int_{[-\pi, \pi]^d} \frac{\sin(t f(\mathbf{k}))}{\sin(f(\mathbf{k}))} e^{i t \mathbf{...
Ko Hey's user avatar
  • 81
5 votes
1 answer
220 views

How big is the class of all closed range bounded linear operator?

Let $X$ and $Y$ be Banach spaces and let $CR(X,Y)$ denote the set $B(X,Y)$ of all bounded linear maps from $X$ to $Y$ with $T(X)$ closed in $Y$. Certainly $CR(X,Y)$ is not open in $B(X,Y)$ as given ...
Anupam's user avatar
  • 585
2 votes
1 answer
179 views

Reference request: Parabolic Schauder estimates for the heat equation with $f \in L^\infty$

Let us consider the heat equation $$\partial_t u - \Delta u = f(x, t) \quad \text{in }Q_R $$ where $Q_R = B_R \times (-R^2,0].$ I would like to know the kind of regularity we should expect of $u$ if ...
Falcon's user avatar
  • 452

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