**-4**

votes

**0**answers

65 views

### Two problems in functional analysis [on hold]

Let $f$ be linear functional on Banach space $B$ and $ker f$ is closed subspace of $B$, prove that $f$ is a bounded linear functional.
Let $\{e_n\}$ be an orthonormal basis of Hilbert space H. T is ...

**0**

votes

**0**answers

32 views

### Caffarelli-Silvestre extension definition of fractional Laplace-Beltrami on hypersurface

Let $\Gamma \subset \mathbb{R}^{n+1}$ be a $n$-dimensional (closed) $C^k$-hypersurface. Consider the problem
$$\nabla_\Gamma \cdot (y^{1-2s}\nabla_\Gamma u)(x,y) =0 \quad \text{for $(x,y) \in \Gamma ...

**0**

votes

**1**answer

141 views

### How to solve this differential equation with an infinite sum?

I would like to find solutions of the following differential equation:
$ \sum_{1}^{\infty} a_n f(nx) + f''(x)+ x^2 f(x)=\lambda f(x)$
For example in space of function from $\mathbb R^*$ to $\mathbb ...

**2**

votes

**0**answers

38 views

### Existence of solution to weak form of linear equation with boundary integral (parabolic PDE)

Let $W(0,T) := \{ u \in L^2(0,T;H^{\frac 12}(\partial\Omega)) \mid u_t \in L^2(0,T;H^{-\frac{1}{2}}(\partial\Omega))\}$. Let $\gamma$ and $\xi$ denote the trace map and its right inverse.
Does there ...

**6**

votes

**1**answer

170 views

### Diagonalization of the matrix $(1/(i+j+\rm{const}))_{i,j}$

Consider the following infinite matrix: $A_{i,j}=\frac1{i+j+\gamma}$, $0\leq i,j<\infty$, $\gamma>0$ is a constant. Is it known how to diagonalize $A$, or, say, calculate $(I+tA)^{-1}$ for ...

**1**

vote

**1**answer

53 views

### Image of (right) inverse trace map $\xi\colon H^{\frac 12}(\partial\Omega) \to H^1(\Omega)$ dense in $H^1(\Omega)$?

Let $\gamma\colon H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$ be the linear trace map which has a right continuous inverse $\xi\colon H^{\frac 12}(\partial\Omega) \to H^1(\Omega)$.
Is the image of ...

**-4**

votes

**0**answers

41 views

### Weak convergence in Lp [on hold]

Got a little problem with this ex. I could use some help.
Let $U := \Pi_{i=1}^d(a_i, b_i) \subset \mathbb{R}$ ($a_i < b_i$ for each $i$) and let $f \in L^p(U)$ for some $1<p<+\infty$. Let us ...

**0**

votes

**0**answers

84 views

### A consequenc of a Lie group act on a Riemannian manifold by isometry

I am learning differential geometry for using this topic in my research. I am stuck to prove following Result which I got in a article.
Formulation:
Let $ f: [0, 1]\rightarrow \mathbb{R}^2$ be a ...

**2**

votes

**2**answers

97 views

### Uniformly bounded operator family and pointwise convergence

Let $1 \leq p < \infty$ be fixed and let $\Omega \subseteq \mathbb{R}^n$ be open. Let $(Q_n)_{n \in \mathbb{N}}$ be a uniformly bounded family of operators on $L^p(\Omega)$, i.e. there exists ...

**0**

votes

**1**answer

92 views

### Complete solution set of a Convolutional Equation?

Here is a problem that am I stuck and I appreciate any help. In essence, I am trying to show that the only solutions for the described problem are the ones provided below. Best..
Setup: In what ...

**0**

votes

**0**answers

51 views

### Reference : Special case of Banach-valued function integration by parts

Let $u \in L^p (0,T; L^1(\Omega))$ with $\partial_t u \in L^p(0,T; L^1(\Omega))$ and $v \in L^q (0,T; L^\infty(\Omega))$ with $\partial_t u \in L^q(0,T; L^\infty(\Omega))$ (with $1/p+1/q=1$ and $p \in ...

**-4**

votes

**0**answers

109 views

### show that L(X,Y)banach then Y banach

Let {Xα}α∈A be a collection of Banach spaces. It is easy to show that P={(xα):supα∥xα∥<∞} with ∥(xα)∥=supα∥xα∥ is a banach space.
If the indexing set A is finite, then it is easy to show that P ...

**0**

votes

**0**answers

90 views

### Roots in the solution

It is known that for a one-dimensional self-adjoint operator with periodic boundary conditions, the number of roots is directly related to the eigenstate this eigenfunction belongs to.
Now it is ...

**0**

votes

**0**answers

49 views

### Schoenberg correspondence on $L^p$

Schoenberg correspondence states that $\psi: \mathbb R^d\longrightarrow \mathbb C$ is conditionally positive definite and hermitian if and only if $e^{t\psi}$ is positive definite for each $t>0$. ...

**1**

vote

**1**answer

92 views

### Rademacher type of a Banach space is always less than or equal to 2

Before I ask my question I will provide a brief introduction.
I came across the notion of Rademacher type while reading Assaf Naor's article An introduction to the Ribe program, which can be found ...

**1**

vote

**0**answers

59 views

### Better version of “Monotonicity methods in Hilbert spaces and some applications to nonlinear PDEs..”

I am asking whether any one knows of a better source for the text
Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential
equations by H. Brezis
which I ...

**-2**

votes

**0**answers

146 views

### Why the definition of continuity is not reversed? [closed]

I am wondering why continuity is not defined in the reverse order. Take the definition of continuity in topology for example, its definition is defined as:
mapping of a topological space $(X, T_X)$ ...

**5**

votes

**0**answers

136 views

### Operator arithmetic-harmonic mean inequality with operator-valued weights

Let $\Lambda_1,\dots,\Lambda_n$ be strictly positive definite operators in the Euclidean space $\mathbb{R}^d$. By an operator arithmetic-harmonic mean inequality with weights $\Lambda_i$ I mean the ...

**2**

votes

**0**answers

109 views

### Classify spaces that make extension theorems hold

Recall a Polish space is a completely metrizable separable space.
Say a Polish space $Y$ is a terminal space if for any Polish space $X$ and any closed $C \subseteq X$, one can extend a continuous ...

**0**

votes

**2**answers

82 views

### Book and Papers for properties of uniformly convex and locally uniformly convex and strictly convex Banach spaces.

I am looking for reference books and research articles which cover analysis of uniformly convex and locally uniformly convex and strictly convex Banach spaces.

**7**

votes

**1**answer

300 views

### Is there a simple direct proof of the Open Mapping Theorem from the Uniform Boundedness Theorem?

The Open Mapping Theorem, the Bounded Inverse Theorem, and the Closed Graph Theorem are equivalent theorems in that any can be easily obtained from any other. The Closed Graph Theorem also easily ...

**2**

votes

**0**answers

81 views

### Speed of Approach to Invariant Measure

Let $X_t$ represent a continuous-time Markov process on $\mathbb{R}^d$, say a diffusion with locally Lipschitz coefficients. Suppose that there exists a unique invariant measure $\mu$ on the space, ...

**0**

votes

**0**answers

73 views

### The singular value of $F(\theta)=\sin\theta\int_{-a}^{a}e^{-ikz\cos\theta}f(z)dz.$ [closed]

This is the continuation of a previous question Inverse problem to solve out current in the radiation problem? Suppose that a current $f(x)\in L^2(\mathbb{R})$ is compactly supported in $[-a,a]$, ...

**6**

votes

**1**answer

160 views

### Domains of raising and lowering operators in QM

I should add that what is known in physics as SUSY QM, is often called the Crum-Darboux method in Mathematics, so don't be confused about this.
Let $H : \operatorname{dom}(H) \subset L^2(\Omega) ...

**2**

votes

**1**answer

111 views

### Well-posedness of heat equation with distributional right hand side

The question is about well-posedness of heat equation
$$
\frac{\partial\Theta}{\partial t}=\alpha^2\Delta\Theta+p(t)\delta(x-u(t))\delta(y-v(t)),~~ (x,y,t)\in\Omega\times[0,T],
$$
subjected to ...

**0**

votes

**1**answer

50 views

### Galerkin Projection on Integral Operators

I am looking at a research paper that mentions integral operators (which in this case is brought up in reference to shading equations that are integral operators) and it says that we can create a ...

**0**

votes

**0**answers

53 views

### Implicit function theorem on boundary points

I have the following examples:
(1) $xy-1=0$ with $x\ge 0$. By the implicit function theorem, we can solve when $x\in(0, \infty)$. Here on the boundary we have $y=\frac{1}{x}\rightarrow \infty$ as ...

**2**

votes

**1**answer

124 views

### Eigenfunctions of an infinite summation operator

I would like to find ALL eigenfunctions to the operator, for $f$ a real function on R+*:
$f \rightarrow \sum_{1}^{\infty} f(nx)$
So to find $f$ such that: $\sum_{1}^{\infty} f(nx) = \lambda f(x)$
...

**0**

votes

**0**answers

115 views

### $Ax=b$ in a function space (again)

Let
$X$ be compact Hausdorff topological space,
$C(X)$ denote the algebra of complex-valued continuous functions on $X$,
$b\in \mathbb{C}^m$,
$\mathbf{A}\in C(X)^{m\times n}$,
Let ${\mathbb{C}}^n$ ...

**3**

votes

**1**answer

115 views

### $\lVert u\rVert_{W^{2,p}}$ is bounded above by $\lVert \Delta_p u\rVert_{L^2}$ for $u \in W^{1,p}_0 \cap W^{2,p}$?

I work on a bounded domain in $\mathbb{R}^n$ and let $p \geq 2$ and the operator $\Delta_p u = \nabla \cdot (|\nabla u |^{p-2}\nabla u)$.
Does the following inequality (or something similar hold) for ...

**1**

vote

**0**answers

73 views

### Free Endpoint of Minimization Problem

Consider the following minimization problem $$\inf \left\{ \int\limits_{-\infty}^0 \left[ (\psi')^2 + m(y)(\psi - F)^2 \right]\; : \; \psi \in H^1(\left(-\infty,0\right]) \right\}$$ where $m(y) > ...

**0**

votes

**0**answers

63 views

### Dipole Transition Integrals - Acceleration Form, What's Wrong?

I should have posted this question in a physics forum, but I think by posting in MathOverflow I may get more responses.
The following question may sound stupid, since I'm sure I was wrong somewhere, ...

**2**

votes

**1**answer

246 views

### Banach space of discontinuous functions(Killing continuous functions)

Edit: According to the comment of Prof. Majer, I revise the question:
For a metric space $X$, we put $A=\{f:X\to \mathbb{C}\mid \text{f is bounded}\}$. We define two semi norm on $A$
...

**2**

votes

**1**answer

197 views

### A commutative Banach algebra with an abundance of discountinuous functions

Let $A$ be the algebra of all bounded functions from $[0,\;1]$ to $\mathbb{C}$.
For $f\in A,\;$ $\omega_{f}$ is the standard oscillation function.. Each of the following two (equivalent) norms ...

**3**

votes

**1**answer

239 views

### When $C(X)$ is an injective $C(X)$-module?

It is an old question if every injective Banach space is isomorphic as Banach space to $C(X)$-space.
I would like to know if the weakened module version of this question is answered. More precisely: ...

**0**

votes

**0**answers

23 views

### continuous extensions of concave functions

Let $N$ be a lattice. For a ring R we denote $N_R := N \otimes R$. My question is the following:
Does a continuous and concave function
\begin{eqnarray*}
f: N_{\mathbb{Q}} \to \mathbb{R}
...

**3**

votes

**0**answers

87 views

### Continuously dependent on parameters [closed]

How do we check whether the solution is continuouly dependent on parameters?
Let $\Omega$ be a domain with smooth boundary. Say $f$ and $h$ are smooth. Assume that for each $\theta\in (0, 1]$, the ...

**4**

votes

**1**answer

123 views

### Can one show that the dual of a quasi-Banach space separates points without explicitly identifying the dual?

I'm interested in a question regarding the identification of some duals of quasi-Banach spaces.
However, I'm not familiar with the quasi-Banach literature, so I'm hoping somebody can point me in the ...

**0**

votes

**1**answer

146 views

### $Ax=b$ in a function space

Let
$X$ be compact Hausdorff topological space,
$C(X)$ denote the algebra of complex-valued continuous functions on $X$,
$b\in \mathbb{C}^m$,
$\mathbf{A}\in C(X)^{m\times n}$,
for all $x\in X$, ...

**2**

votes

**1**answer

100 views

### Question regarding to approximate continuity

Given $u\in BV(R^N)$, we say $u$ is approximate continuous at $x$ and the approximation limit is $l\in R$ if
$$ \lim_{r\to 0}\frac{\mathcal{L}^N(B(x,r)\cap \{|u-l|>\epsilon\})}{r^N} =0 $$
for all ...

**5**

votes

**2**answers

135 views

### Compactly supported functions and Sobolev spaces on manifolds

It is well-known that if a complete Riemannian manifold has bounded curvature and injectivity radius bounded away from zero, then the space $C^\infty_c(M)$ is dense in the Sobolev spaces $W^{k, p}(M)$ ...

**0**

votes

**0**answers

69 views

### Inverse problem to solve out current in the radiation problem

This is the radiation problem, but more like a analysis one. Suppose the current $f(x)\in L^2(\mathbb{R})$ has a compact support in $[-a,a]$. And the frequency $F(\theta)$ is given by ...

**1**

vote

**0**answers

154 views

### Separability of the space $C(C[0, 1], \mathbb{R})$

Let $E=C([0, 1])$ be the space of all real-valued continuous functions on $[0, 1]$, equipped with the uniform norm. $C(E)$ stand for the continuous real-valued functions on $E$.
I am wondering that ...

**10**

votes

**2**answers

461 views

### Banach space modulo a one-dimensional subspace =?

My question is the following:
Given an infinite dimensional Banach space $E$ and a one-dimensional linear subspace $F\subset E$. It is well-known that this one-dimensional linear subspace is closed ...

**0**

votes

**1**answer

117 views

### Equivalence of two definitions of Sobolev spaces

Good morning,
I am looking for a reference about the following fact that seems to be folklore. Define the Sobolev (Beppo Levi?) space
$$
D^{1,p}(\mathbb{R}^N) = \left\{ u \in L^{p^*}(\mathbb{R}^N) ...

**0**

votes

**0**answers

62 views

### faithful action of Hecke algebra

Let $G$ be a connected reductive group split over a number field $F$, $\mathbb{A}$ the adeles.
Let $v$ be a finite place and $\mathcal{H}_{v}$ the spherical hecke algebra at palce $v$.
...

**3**

votes

**2**answers

106 views

### Reconstructing density from integrals along specific manifolds

Let $\Phi_t : \mathbb R^n \to \mathbb R^n$ be the time-$t$-map associated to an ODE $\dot{x}=F(x)$ and let $H: \mathbb R^n \to \mathbb R$. Let $F$ and $H$ be sufficiently smooth (e.g. $C^k$ or ...

**7**

votes

**1**answer

235 views

### Bounded operator on a normed space with empty spectrum

A bounded operator acting on a complex Banach space has non-empty spectrum, and the proof of this fact uses the completeness of the space.
Is there any example of bounded operator acting on a ...

**0**

votes

**0**answers

78 views

### Defining density of a random function using Radon-Nikodym Theorem

Let $(\Omega,\mathbb{F},P)$ be a probability space and $E$ be an infinite dimensional Banach space and $\mathbb{B}$ be the $\sigma$-algebra of Borel subset of $E$.
Let $X$ be random function defined ...

**1**

vote

**0**answers

92 views

### Lebesgue point and regularity of functions

A known theorem says that for $f \in L_{loc}^1(\mathbb{R}^d)$, almost every point is a Lebesgue point.
I know too a theorem saying that for $f \in W_{loc}^{1,p}(\mathbb{R}^d)$ , every point is a ...