**0**

votes

**1**answer

65 views

### Existence of a projection operator onto subspace of Hilbert space

Let $V \subset H$ be Hilbert spaces with a continuous and dense imbedding. Let $\{w_j\}_j \subset V$ be a basis of $V$ and of $H$ (so finite linear combinitions are dense) which is not orthogonal. ...

**5**

votes

**1**answer

52 views

### Decay of solutions to Schrodinger equation with local minimum in potential

Consider the one-dimensional Schrodinger operator on the real line $\mathbb{R}$ given by
$$ L = - \partial_x^2 + V $$
where $V$ is a potential with the following properties:
$V$ is non-negative, ...

**0**

votes

**0**answers

40 views

### About expectation norms on graphs

Let $S \subseteq V$ of a $d-$regular graph $G$ such that $\mu = \frac{\vert S \vert }{\vert V \vert } $. Let $A$ be the adjacency matrix of the graph. Then define the quantity $\phi(S)= ...

**6**

votes

**2**answers

229 views

### Relating different topologies on $C^{\infty}_c(M)$

This is somehow connected to this question.
I can think of at least four topologies to put on $C_c(M)$:
Topologize $C^{\infty}_c(M)\subseteq C^{\infty}(M)$ as a subspace with the weak Whitney ...

**1**

vote

**0**answers

45 views

### Find $U \in H^1(\Omega \times (0,\infty))$ such that $\nabla E(u-\bar u)\nabla U \geq 0?$ (PDE harmonic extension)

Let $\Omega$ be a bounded smooth domain. Given $u \in H^{\frac 12}(\Omega)$ with mean value $\bar u = 0$, let $Eu = v \in H^1(\Omega \times (0,\infty))$ solve
$$\int_0^\infty\int_\Omega \nabla v\nabla ...

**0**

votes

**0**answers

81 views

### Reformulation of a theorem for special case [on hold]

This question is about some theorems in the book "Analysis Now" from Pedersen.
In particular Proposition $5.3.2$ and Theorem $5.3.3$. Here one has a essential $*$-isomorphism of some algebra $L(X)$ ...

**0**

votes

**0**answers

22 views

### Uniform bound in Faedo-Galerkin method with time-dependent weight in inner product

Let $v_j$ be an orthonormal basis for $V=H^1(\Omega) \subset L^2(\Omega)$ which is orthogonal in $L^2(\Omega)$.
Let $w:[0,T]\times\Omega \to \mathbb{R}$ be a time-dependent weight which is smoooth ...

**5**

votes

**1**answer

177 views

### $\mathcal S(\mathbb R^n) \hat \otimes_\pi \mathcal S(\mathbb R^m) \simeq \mathcal S(\mathbb R^{n+m})$?

If $S(\mathbb R^n)$ is the Scwartz space of smooth rapidly decaying functions equipped with the topology generated by the family of semi-norms
$$\mathcal N_p (\varphi)= \sum_{|\alpha|, |\beta| \leq p} ...

**6**

votes

**0**answers

98 views

### Max min of functionals

I have an interesting question which I believe was probably already studied, but I could not find anything. Let $n, m \geq 1$ be fixed. Suppose that $|| \cdot ||$ is a norm in $\mathbb{R}^n$ and $f_1, ...

**1**

vote

**0**answers

62 views

### Weak topology on subsets of a Hilbert Space

I have few questions about the subsets of a (for example) Hilbert Space endowed with the weak topology. Let $E$ be such subset.
When is the norm a continuous function on $E$?
This happens for ...

**0**

votes

**0**answers

37 views

### Approximation property of Fréchet if range is restricted to an embedded Hilbert space

Let $W$ be a separable Fréchet space, and $H\subset W$ be a separable Hilbert space that is continuously embedded (equivalently, the topology of $H$ is stronger than the subspace topology generated by ...

**4**

votes

**1**answer

265 views

### Is the conditional expectation a contraction in weak $\mathbb L^p$ spaces?

Let $(\Omega,\mathcal F,\mu)$ be a probability space. It is well-known that if $\mathcal A$ is a sub-$\sigma$-algebra of $\mathcal F$, $p\geqslant 1$ and $X$ is an element of $\mathbb L^p$ which takes ...

**15**

votes

**2**answers

413 views

### Ultraweak topology on B(X): Is the map X\otimes X* -> B(X)* isometric?

Let $X$ be a Banach space. Consider the map
$$
\alpha\colon X\hat{\otimes} X^* \to B(X)^*,
$$
defined one simple tensors as
$$
\alpha(\xi\otimes\eta)(a) = \eta(a(\xi)).\quad (\xi\in X, \eta\in X^*, ...

**44**

votes

**4**answers

3k views

### Did Gelfand's theory of commutative Banach algebras influence algebraic geometers?

Guillemin and Sternberg wrote the following in 1987 in a short article called "Some remarks on I.M. Gelfand's works" accompanying Gelfand's Collected Papers, Volume I:
The theory of commutative ...

**2**

votes

**1**answer

322 views

### Laplacian with singular potential

Let $S$ be a $2$-dimensional sphere. Let $p$ be a point in $S$. Let $L$ be a second order elliptic partial differential operator with smooth coefficients defined over the complement of $p$. Near $p$, ...

**1**

vote

**1**answer

103 views

### Bounded-open topology vs norm on $L\left(X,Y\right)$

In general topology there is two ways of introducing a topology on the space of (continuous) maps between, say, metric spaces: set-open topology and uniform topology (it is a uniformity of uniform ...

**3**

votes

**1**answer

92 views

### Weak convergence in $W^{1,p}_0$

Note from the answerer : this question stems from this article.
I ask this question in http://math.stackexchange.com/questions/1206617
I have a bounded sequence $(u_n)$ from $W^{1,p}_0(\Omega)$ so ...

**3**

votes

**2**answers

169 views

### distributional Hessian for semiconvex functions on non-smooth manifolds

Let $f:R^n \to R$ be convex. Then there exist signed Radon measures $\mu^{ij}=\mu^{ji}$ such that
$$
\int_{R^n} f \frac{\partial^2 \varphi}{\partial x_i \partial x_j} dx= \int_{R^n} \varphi d\mu^{ij} ...

**0**

votes

**0**answers

69 views

### absolutely continuous of two probability measures

Suppose $X_t$ satisfies
$$X_t=\int_0^t b(X_s)ds+ L_t,\quad t\in[0,1]$$ where $L_t, t\in[0,1]$ is a $\alpha-$stable process. Let $P_L$ be the law of $L$, $P_X$ be the law of $X$. ($P_L, P_X$ are ...

**1**

vote

**1**answer

137 views

### How to prove that $(1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$) for $x\ll 1$ in this specific case

After multiple plots I noticed that function $h(x)= (1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$), for $x\ll 1$ (specifically $0<x<0.1$) and ...

**1**

vote

**1**answer

77 views

### A differential inequality and a special value

Let $G \colon [0,1] \to [0,1]$ be a monotonically decreasing function with $G(0) = 1$ and $G(1) = 0$. Suppose that $G$ is differentiable infinitely many times, and that: $$G(x)G''(X) \leq ...

**-2**

votes

**0**answers

15 views

### Approximate point spectrum is complement of set of points of regular type [migrated]

I have a question concerning the approximate point spectrum of a closed linear operator. I need to show that the approximate point spectrum is the complement of the set of points of regular type, ...

**4**

votes

**3**answers

484 views

### reflexive banach space

I want to ask this non-expert question:
What does it mean geometrically for a Banach space to be reflexive?
Well, we could say a Banach space is reflexive iff unit ball is weakly compact. Or some ...

**-3**

votes

**0**answers

32 views

### Averages of bounded function [migrated]

For a continuous $f\colon\mathbb{C}\longrightarrow[0,1]$, what could be said about $\mathcal{F}=\{f_r\colon r>0\}$ where $$\forall\,r>0,\,\forall\,z\!\in\!\mathbb{C},\quad ...

**1**

vote

**1**answer

83 views

### About the upper bound on the roots of the matching polynomial

Heilman and Lieb had proven that if a graph had $d$ as its maximum vertex degree then the roots of the matching polynomial are bounded from above by $2\sqrt{d-1}$.
Is there a modern exposition of ...

**-2**

votes

**0**answers

39 views

### periodic function satisfy the condition f’’(x)f(x)>0 at -inf < x < +inf? [closed]

Can a periodic function f(x) satisfy the condition f’’(x)f(x)>0 at -inf < x < +inf?

**-3**

votes

**1**answer

72 views

### Can functional invariants of dynamical systems be used in data science (or parameter identification)? [closed]

Given a functional of the form:
$$
F[x]=\int^{t}_{0} \mathcal{L}(x^{(n)},x^{(n-1)},...,x,\tau)\,\text{d}\tau+g(x(t))
$$
Where $x$ is in $\mathbb{R}^{m}$ and is in ...

**1**

vote

**0**answers

106 views

### Dual space of $l^p(\mathbb{Z},X)$

Let $X$ be a Banach space, $p \in [1,\infty)$ and $l^p(\mathbb{Z},X)$ the usual sequence space taking values in $X$. Is it always true that $(l^p(\mathbb{Z},X))^* = l^q(\mathbb{Z},X^*)$ and ...

**-1**

votes

**1**answer

114 views

### Analysis of Sobolev spaces [closed]

I just wanted to know wthether the following is OK or not.
Let $X$ be $H_0^1(\Omega)\bigcap L^{\infty}(\Omega)$, thought of as a subspace of $H^1_0(\Omega)$ and endowed solely with the usual $H^1$ ...

**3**

votes

**4**answers

579 views

### Non-separable Banach space

The vector space $C_b(\mathbb R)$ of bounded continuous functions on $\mathbb R$ is non-separable: it is possible to produce a direct proof of this fact, mimicking the standard proof for the ...

**0**

votes

**0**answers

169 views

### Theorems about matrices with entries from $0,1,-1$? [closed]

Consider matrices which are of the form $\left [ \begin{matrix}
0 && A\\
A^T && 0 \\
...

**1**

vote

**1**answer

169 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 ...

**2**

votes

**1**answer

128 views

### Non-closability of an operator

Let $a$ be a positive continuous function nowhere differentiable on $[0,1]$. The operator $T$ in $H:=L^2(0,1)\oplus L^2(0,1)$ defined by
$$T(u_1,u_2) := (u_1' + au_2',0)$$
on $\textrm{Dom} \,T := ...

**2**

votes

**0**answers

73 views

### On uniform or simple convergence of Poisson Summation formula

Under good conditions on an even function $f(x)$ we have the Poisson Summation formula ($x>0$):
$$f(0) + 2 \sum\limits_{n =1}^{\infty} f(nx)= \frac{1}{x} \left( \hat{f}(0) + 2 \sum\limits_{n ...

**7**

votes

**2**answers

555 views

### Decomposition of an integral operator into a composition

I've been musing about the following question for a while now. Given an integral operator $G$ defined by
$$ (Gf)(x) = \int_0^1 G(x,u) f(u)\,du $$
Is it possible to decompose this into two separate ...

**2**

votes

**0**answers

94 views

### Sobolev space for manifold with boundary

For an compact manifold $M$ without boundary, we consider the eigenfunctions $(f_1,f_2,\ldots)$ of some elliptic operator (e.g $\Delta$) with eigenvalue $\lambda_{1},\lambda_{2},\ldots$. To define ...

**0**

votes

**1**answer

69 views

### Motivating the Bessel translation operator

In a paper I am reading on the Hankel transform (this paper to be exact), I've come across a somewhat peculiar definition for a generalized translation operator. The operator is designed with a ...

**1**

vote

**0**answers

62 views

### Da Prato's notion of Symmetric Operator

For anyone who's familiar with G. Da Prato's books on infinite dimensional analysis, I was wondering if someone could clarify something. In, for instance, "An Introduction to Infinite Dimensional ...

**0**

votes

**1**answer

135 views

### Functional Calculus and Fredholm index

Let $-\Delta: W^{2,2} \subset L^2(\mathbb{S}^2) \rightarrow L^2(\mathbb{S}^2)$. Then it is "easy" to show that $-\Delta $ is self-adjoint. Now, I am looking for closed operators $T$ and $T^*$ of order ...

**2**

votes

**0**answers

98 views

### Infinite sums with Mobius Inversion : can we have uniform convergence of inversion formula?

My question is on Mobius inversion formula convergence/properties when used with infinite sums of function.
Lets consider (on $\mathbb{R}^{+}$):
$$S(x)= \sum\limits_{n=1}^{\infty} f(nx)$$
We call ...

**1**

vote

**1**answer

106 views

### Is a Fréchet Montel space distinguished?

Based on a couple of references, it seems that the answer is yes, see for example
Boneta-Dierolf, 1992 and Bierstedt-Bonet, 1989.
However, from a comment to the answer of this MO question, I infer ...

**2**

votes

**1**answer

102 views

### Nuclearity properties of Integral operators with Schwartz type kernels

Consider an integral operator $$T:L^2({\mathbb R}^n)\to L^2({\mathbb R}^n),\qquad (Tf)(x)=\int_{\mathbb R^n} dy\,K(x,y)f(y),$$ with Schwartz type kernel $K\in{\mathscr S}({\mathbb R}^{2n})$ (smooth ...

**3**

votes

**1**answer

53 views

### Relatively compact sets in Ky Fan metric space

Let $(\Omega,P,\mathcal{F})$ be a probability space. $X$, $Y$ are two random variables. The Ky Fan metric defined as: $d_F(X,Y)=\inf\{\epsilon: P(|X-Y|> \epsilon)<\epsilon\}$ (or $d'_F(X,Y)=E ...

**3**

votes

**0**answers

62 views

### Is Wiener's Tauberian theorem true in Wiener space?

Let $\gamma$ be the standard product Gaussian measure in $\mathbb{R}^\infty$, and let $\mu$ be a finite variation measure, not necessarily positive, such that $\mu \ll \gamma$.
Is the following true?
...

**13**

votes

**2**answers

1k views

### Dual of the space of Hölder continuous functions?

Let $X=C^{\alpha}(\Omega,\mathbb{R})$ be the space of Hölder continuous functions. What is its dual?

**7**

votes

**0**answers

157 views

### Lipschitz-free spaces of $\mathbb R^n$

We define
$$
\text{Lip}_0(\mathbb R^n)=\{f:\mathbb R^n\rightarrow \mathbb R, \text{such that $f(0)=0$ and }
\sup_{x\not=y}\frac{\vert f(x)-f(y)\vert}{\vert x-y\vert}<+\infty.
\}
$$
It is well-known ...

**1**

vote

**1**answer

264 views

### Euler-Lagrange Equation and “Eigen Value ”

I posted this question on Math.SE, but I could not get any help.
The eigenvalue $\lambda(t)$ is characterised as the minimum of the Rayleigh quotient (where $t$ is a scalar variable)
...

**1**

vote

**0**answers

90 views

### Topological properties of space of Radon measures

Let $M$ denote the space of signed unbounded Radon measures on $\mathbb{R}$ as is defined by Bourbaki, i.e. $M$ is the dual of $C_c$ where $C_c$ is the space of continuous functions on $\mathbb{R}$ ...

**2**

votes

**2**answers

88 views

### Infinite direct sum of $l_2^{(n_k)}$ contains a complemented isometric copy of $l_2$

How do I show that for any increasing sequence $(n_k) \subseteq \mathbb{N}$, the space $\left( \oplus _{k=1} ^\infty l_2 ^{(n_k)} \right) _\infty$ contains a complemented isometric copy of $l_2$?

**1**

vote

**1**answer

85 views

### Orthogonal compact operators on an infinite dimensional Hilbert space [closed]

How do I show that when $H$ is an infinite-dimensional Hilbert space we can find two compact positive operators $u,v$ with infinite dimensional image and $u \perp v$?
This statement can be found at ...