# Questions tagged [fa.functional-analysis]

Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

5,467 questions

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### Approximation of functions by tensor products

Given a function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$, can we find a sequence of functions $f_n$ of the form $f_n(x,y)=\sum_{i=1}^ng_i(x)h_i(...

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47 views

### Approximation of functions in $L^p(R^d;L^\infty)$

Assume that the function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$. Can we find a class of functions $f_n\in C_b^2(R^d;L^\infty(B_R))$ such that
$$...

**3**

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**1**answer

48 views

### Question on relation between a parabolic sobolev space and a sobolev bochner space

For parabolic sobolev spaces I follow the following definition:
According to this definition, we have that $W^{1,1,2}(I \times \Omega)=L^2(I; W^{1,2}(\Omega)) \cap W^{1,2}(I; W^{-1,2}(\Omega))$
...

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66 views

### Tensor product of compact operators on Banach modules

Let $A$ and $B$ be Banach algebras. Consider a right Banach $A$-module, $E$, and a right Banach $B$-module, $F$, as well as a Banach algebra morphism $\pi\colon A\to\mathcal L_B(F)$ into the bounded $...

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**1**answer

262 views

### Where to find the proof of this property?

I am doing some exercises in the analytic and there is a problem as following:
``Let $\{f_n\}_{n \in \mathbb{ N}}$'' to be a positive sequence such that:
$\sum\limits_{n=1}^{+\infty} f_n = 1$.
$\...

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48 views

### What is the relationship (mapping) from a reciprocal function 1/r to a exponential function exp(-r)? [on hold]

The mathematical problem:
Consider the mapping from $r$ to $u$.
for large $r$ the theory suggests a formulation like $u=a_1 e^{-a_2 r}$, which means that the function decay exponentially.
for ...

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96 views

### Affine algebraic variety as a set of common zeroes of holomorphic functions on ${\mathbb C}^n$

Let $V$ be an affine algebraic variety in ${\mathbb C}^n$, i.e. a set of common zeroes of some set $S$ of polynomials on ${\mathbb C}^n$:
$$
V=\{z\in {\mathbb C}^n:\ \forall p\in S\quad p(z)=0\}.
$$
...

**3**

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**1**answer

61 views

### Mapping inclusion theorem for the numerical range

We denote the numerical range of a complex square matrix $A \in \mathbb{C}^{n\times n}$ by $W(A)$.
Let $A \in \mathbb{C}^{n\times n}$ and let $f: \mathbb{C} \to \mathbb{C}$ be, say, an entire ...

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**1**answer

82 views

### Complemented subspaces of Lorentz sequence spaces?

Let $d(\textbf{w},p)$, $1\leq p<\infty$, denote the Lorentz sequence space, where $\textbf{w}=(w_n)_{n=1}^\infty\in c_0\setminus\ell_1$ is a normalized decreasing weight.
Is there very much known ...

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64 views

### Dual Lorentz spaces

MO seems the perfect place to ask for the following question. Denote the Lorentz spaces on an arbitrary measure space $(E,\mu)$ by $L^{p,q}=L^{p,q}(E,\mu)$, and by $p'$ the conjugate index of $p$.
...

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**1**answer

190 views

### Quasinilpotent , non-compact operators

If $X$ is a separable Banach space and $(\epsilon_n)\downarrow 0$, can we find a quasinilpotent, non-compact operator on $X$ such that $||T^n||^{1/n}<\epsilon_n$ for all $n$? I suspect the answer ...

**3**

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**3**answers

140 views

### Existence of solution to linear fractional equation

We consider the equation
$$ \sum_{j=1}^n \frac{\lambda_j}{x-x_j} =i$$
where $\lambda_j>0$ and $x_j$ are real distinct numbers.
I want to show that if $\lambda_k$ is small compared to the ...

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68 views

### Duality mapping of a the space of continuous functions? [closed]

The duality map $J$ from a Banach space $Y$ to its dual $Y^{*}$ is the multi-valued operator defined by: $J(y)=\{\phi\in Y^{*}:\, \left< y,\phi\right >=\Vert y\Vert^{2}=\Vert \phi \Vert^{2}\},...

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147 views

+100

### Approximation of a compactly supported function by Gaussians

Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function whose support is a closed interval, e.g. $\text{supp}(f)=[a,b]$. Then $f$ can be approximated (e.g. in $L^2$) by a linear combination of Gaussian ...

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120 views

+100

### Open Questions about Wasserstein Space and PDE

While working on my thesis, I encountered the idea of OMT and started reading some more (like Villani's book). In particular, I came across a PhD thesis by Martial Agueh. I thought it was interesting ...

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**1**answer

162 views

### Do analytic functionals form a cosheaf?

Let $X$ be a complex-analytic manifold.
Consider the sheaf of holomorphic functions $\mathcal{O}_X$ as a sheaf with values in the category of locally convex vector spaces. For $U\subseteq X$ open, we ...

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75 views

### Lower semicontinuous and convex envelope

L.Ambrosio, in paper [1] writes:
Let $g:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$ be a function (...)
for every $s\in\mathbb{R}$, $z\in\mathbb{R}^n$; we denote, with a slight abuse ...

**3**

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**1**answer

104 views

### Equivalent notion of approximate differentiability

Is it true that the definition of approximate differentiability presented here of a function $f: \mathbb{R}^N \to \mathbb{R}$ is equivalent to the following one?
$$\lim_{r \to 0} \rlap{-}\!\!\int_{...

**1**

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**1**answer

79 views

### Two PDE for one unknown?

Let $x \in (0,L)$, $t \in (0,T)$, and let $f_1 = f_1(x,t) \in \mathbb{R}$, $f_2 = f_2(x,t) \in \mathbb{R}$, $u^0 = u^0(x) \in \mathbb{R}$ and $g= g(t) \in \mathbb{R}$ be continuous functions.
My ...

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59 views

### Functional characterization of local correlation matrices?

Definition: A matrix $C\in\mathbb R^{m\times n}$ is local correlation matrix iff there exists real random variables $x_1,\dots,x_m,y_1,\dots,y_n$ defined on a common probability space which takes ...

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**2**answers

272 views

### Properties of heat equation

** I simplified the question: **
On bounded domains, the maximum principle implies that the solution to the heat equation is (strictly) positive, if the initial and boundary data is positive.
I ...

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70 views

### Measurability of the heat semigroup in $L^\infty$

Let $S(t)$ be the $C_0$-semigroup generated by the Laplacian operator with Dirichlet boundary condition in $L^2(\Omega)$, where $\Omega$ is a bounded open subset of $R^n$.
It is known that $S(t)$ ...

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31 views

### Prove that the Bynum space $\ell ^{p, \infty}$ is a uniformly not square space

In the book Measures of Noncompactness in Metric Fixed Point Theory, the authors prove that $\varepsilon_{0}(\ell ^{p, \infty}) = 1$, where $\ell ^{p, \infty} = (\ell ^{p}, \Vert \cdot \Vert _{p, \...

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131 views

### Largest weak(-like) topology with respect to which continuous functions are dense in the space of Borel functions

Let $X$ denote the space of bounded Borel functions $f\colon [0,1] \to \mathbb{R}$. Let $M$ denote the space of finite Borel measures on $[0,1]$. What is the largest family $F \subset M$ such that for ...

**6**

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**1**answer

186 views

### Compactness of set of indicator functions

Let $\chi_A(x)$ denote an indicator function on $A\subset [0,1]$. Consider the set
$$K=\{\chi_A(x): \text{ A is Lebesgue measurable in }[0,1]\}.$$
Is this set compact in $L^\infty(0,1)$ with respect ...

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47 views

### About a class of expectations

Consider being given a $n-$dimensional random vector with a distribution ${\cal D}$, vectors $a \in \mathbb{R}^k$, $\{ b_i \in \mathbb{R}^n \}_{i=1}^k$ and non-linear Lipschitz functions, $f_1,f_2 : \...

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**1**answer

182 views

### Can the characteristic function of a Borel set be approached by a sequence of continuous function through a certain convergence in $L^\infty$?

Want to find $f_n$ a sequence of continuous functions, so that for all Borel regular measure $\mu$, we have
$\int f_n d\mu\rightarrow\int \chi_\Delta d\mu$, as $n$ goes to infinity, where $\Delta$ ...

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34 views

### Is every pair of closed linear subspaces boundedly regular?

In the context of the development of the spectral theory, J. v. Neumann showed that for every pair of closed linear subspaces $M,N$ of a Hilbert space the iterations $(P_M P_N)^nx$, where $P_M$ and $...

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31 views

### Function classes with high Rademacher complexity

My question is two fold,
Is there any general understanding of what makes a function class have high Rademacher complexity? (Sudakov minoration would say that one sufficient condition for a class of ...

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**1**answer

121 views

### Brascamp-Lieb inequalities on the sphere

In the paper [CLL], Carlen, Lieb, and Loss demonstrate a version of the Young inequality on the sphere $S^{N-1}$ in $\mathbb{R}^N$. For positive functions $f_j$ on $[-1,1]$, the following bound holds:...

**2**

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**1**answer

65 views

### Existence of a bounded operator which satisfies the discrete product rule

Is there a bounded self-adjoint operator $H$ acting on $\ell^2(\mathbb{Z})$ such that for all sequences $u,v\in \ell^2(\mathbb{Z})$
$$ H(uv)=(Hu)v+u(Hv)$$
where uv is the pointwise product.
This is ...

**3**

votes

**1**answer

106 views

### Ultraproduct of non-commuative $L^p$-spaces

Let $1<p<\infty.$ Let $I$ be a non-empty set and $\mathcal{U}$ be an ultrafilter over $I.$ Let $M_i$ be von Neumann algebras equipped with normal faithful semifinite traces $\tau_i,$ $i\in I.$ ...

**-1**

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**0**answers

51 views

### How to prove the binary function uniformly boundary?

Assume that $a_1<1$, $a_3<1,$ $a_1+a_2+a_5>1$, $a_3+a_4+a_5>1,$ $a_1+a_2+a_3+a_4+a_5>2.$ For $x,y>0,$ define a fucntion
$$H(u,v)=\frac{u^{\frac{1}{2}}\int_0^{\infty}\int_0^{\infty}\...

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**0**answers

60 views

### Domain of the Generator of a Bessel process

Consider the Bessel Process of index $\nu\in (-1,0)$, or dimension $\delta=2\nu-1$
\begin{align}
\rho_{t}=x+\frac{\delta-1}{2}\int_{0}^{t}\frac{1}{\rho_{s}}\,ds+W_{t}
\end{align}
where $(W_{t})_{t\geq ...

**8**

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**1**answer

245 views

### Laplacian spectrum asymptotics in neck stretching

Let $M$ be a compact Riemannian manifold. Let $S \subset M$ be a smooth hypersurface separating $M$ into two components. Let $g_T$ be a family of Riemannian metric obtained by stretching along $S$, i....

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34 views

### The significant role of dual frames in the progress of Frame theory

For a given frame $\{\zeta_i\}_{i=1}^\infty$, any Bessel sequence $\{\eta_i\}_{i=1}^\infty$ satisfying in the following identity for every $\xi\in H$
$$\xi=\sum_{i=1}^\infty \langle \xi, \eta_i\...

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**1**answer

124 views

### $L^{2}$ Betti number

Let $\tilde{X}$ be a non-compact oriented, Riemannian manifold adimits a smooth metric $\tilde{g}$ on which a discrete group $\Gamma$ of orientation-preserving isometrics acts freely so that the ...

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**1**answer

77 views

### Measure on union of measure spaces and on quotient space

There are two questions about measures bothered me a lot.
Given a set X and a countable covering ${U_i}$ of $X$. Suppose that for each i, there is a measure $m_i$ on $U_i$. Is there a very general ...

**1**

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**1**answer

56 views

### Convergence of probability density function

There are various kinds of (convergence of random variables) but I have never read about convergence of density functions.
Let $X_1, X_2, \dots, X$ be random variables $\Omega \to \mathbb{R}$ and $...

**17**

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**2**answers

701 views

### When is $\sum_{n\in\mathbb Z} f(x+n)$ constant?

A recently asked question (linked here) deals with the remarkable identity
$$ \sum_{n\in\mathbb Z} \mathrm{sinc}(n+x)= \pi,\quad x\in\mathbb R, $$
where $\mathrm{sinc}(x)=\sin(x)/x$.
It is easy ...

**3**

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**2**answers

138 views

### Ultraweak topology in abelian von Neumann algebras

Let $A$ be an abelian von Neumann algebra acting on the (not necessarily separable) Hilbert space $\mathcal{H}$ (with identity $I$). From the Gelfand-Neumark theorem, there is a compact Hausdorff ...

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58 views

### Estimate of $L^1$-norm of semilinear elliptic inequality

Let $\Omega$ be $\mathbb R^n$ or a complete non-compact manifold, we consider
$$\Delta u+f\cdot u+u^2\leq0,$$
where $\Delta$ denotes $-\sum^n_{i=1}\partial^2_{x_i}$ and $f$ is a $C^2$ function such ...

**6**

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124 views

### Can we approximate any eigenvalue of an infinite matrix via eigenvalues of some sequence of submatrices which approximates the matrix?

Let $T:\ell^2\to\ell^2$ be a compact linear operator. Let $[T]=(a_{i,j})_{i,j=1}^{\infty}$ be the representing infinite matrix of $T$ with respect to the canonical base. Let $T_n$ be the finite rank ...

**2**

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**1**answer

78 views

### Discrete dynamical system and bound on norm

Let $z \in \mathbb R\backslash \left\{2 \right\}$ then I would like to understand the following:
Consider the dynamical system with $x_i \in \mathbb C^2:$
$$ x_{i} = \left(\begin{matrix} z &&...

**2**

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**0**answers

70 views

### 1D Schrödinger Equation with Measure-Valued Coefficients

I've been looking at one of the simplest systems I can think of: a one-dimensional infinite square well on $[0,1]$ with Hamiltonian given by the following:
$$\hat{H}=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}...

**4**

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57 views

### Perturbation theory compact operator

Let $K$ be a compact self-adjoint operator on a Hilbert space $H$ such that for some normalized $x \in H$ and $\lambda \in \mathbb C:$
$\Vert Kx-\lambda x \Vert \le \varepsilon.$
It is well-known ...

**1**

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46 views

### Adjoint for a non-densely defined unbounded operator on a Hilbert space

Let $\mathbf{H}$ be a Hilbert space, and $D$ an unbounded densely-defined operator on $\mathbf{H}$. As is well-known, every such operator admits an adjoint, with domain possibly different from that ...

**2**

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64 views

### Off-diagonal estimates for Poisson kernels on manifolds

Let $(M,g)$ be a complete Riemannian manifold, $\Delta$ its Laplace-Beltrami operator and $T_t = (e^{t \Delta})_{t \geq 0}$ the associated heat semigroup. We can define the subordinated Poisson ...

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162 views

### One question about Schrodinger Semigroups-(B. Simon)

This question comes from the paper: B. Simon, Schrodinger Semigroups, Bull. A.M.S., (1982) Vol. 7 (3).
On the Theorem C.3.4(subsolution estimate) of the paper, it says that: Let $Hu=Eu$ and $u\in L^...

**0**

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**0**answers

92 views

### Compact operator

Let $k:[0,1]^2 \to [0,1]$ be a measurable function. Define $K:L^2([0,1])\to L^2([0,1])$ to be the operator:
$$
(Kf)(x) = \int_0^1\int_0^1 f(z) k(x,y) \mathbf{1}_{x\leq z\leq y} \ \mathrm{d}z \mathrm{d}...