Questions tagged [fa.functional-analysis]

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

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40 views

Unbounded subsets of $\ell^+_\infty$ that have non-empty interior and non-trivial unique minimizers

Let $\ell^+_\infty$ be the space of non-negative bounded sequences equipped with the $\sup$ norm topology. Note that the dual of $\ell_\infty$ is the space of finitely additive signed measures on $2^\...
3
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138 views

What do people call functionals on holomorphic functions and on polynomials?

There are four most important functional spaces in analysis: the space $\mathcal{C}(M)$ of continuous functions on a topological space, the space $\mathcal{E}(M)$ of smooth functions on a smooth ...
4
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1answer
163 views

Example: traceless C*-algebra universally generated by projections

Are there examples of a non-zero C*-algebra which is universally generated by finitely many projections (not all commuting) together with a unit and plus necessarily satisfying some additional ...
2
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1answer
96 views

Infinite-dimensional analogue of "positive-negative splitting implies non-degeneracy"

(This question is related to Splitting a space into positive and negative parts but different.) Given a finite-dimensional vector space $V$ over $\mathbb{R}$, what I call a "positive-negative ...
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521 views

Green's function of Cauchy-Riemann operator on Torus

Consider the torus $\mathbb T^2:=\mathbb C/(\mathbb Z+i \mathbb Z)$ and the operator $$ T = (2D_{\bar z}-\lambda)^{-1} $$ on the torus with periodic boundary conditions. This one is well-defined for $\...
4
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1answer
93 views

The symmetric group of positive integers acting on $\ell^\infty(\mathbb{N})^*$

Let $\mathfrak{S}_\mathbb{N}$ be the symmetric group of all positive integers. Let $\ell^\infty(\mathbb{N})^*$ be the dual space of $\ell^\infty(\mathbb{N})$ equipped with weak*-topology. There is a ...
2
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1answer
122 views

$(\iota \otimes f)(X) = 0$ for all $f \in B^*$ implies $X=0$

Let $A$ and $B$ be $C^*$-algebras. Given $f \in B^*$, we can form the right slice map $$\iota \otimes f: A \otimes B \to A: a \otimes b \mapsto af(b)$$ which extends uniquely to a bounded linear map $$...
3
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1answer
140 views

Discrete singular integrals

Let $\{\phi(n)\}_{n\in\mathbb Z}$ be a sequence of complex numbers with the following properties: $\phi(0)=0$ and $|\phi(n)|\leq \frac{C_1}{|n|}$ for all $n\neq 0$ and $C_1>0$ is independent of $n....
3
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1answer
115 views

Is the ball ratio theorem for Radon–Nikodým derivative known for general metric spaces?

Given two non-negative Borel measures $\mu$, $\nu$ on $\mathbb{R}^n$, that are finite on compact sets, such that $\nu\ll\mu$, it is well known that $$\frac{d\nu}{d\mu}(x)= \lim_{\epsilon\to 0} \frac{\...
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1answer
203 views

(Sharp) Bounds on $E(XYZ)$ given all the bivariate marginals

Suppose $X,Y,Z$ are all real-valued random variables. Suppose I know the joint marginal distributions of $(X,Y)$, $(Y,Z)$ and $(X,Z)$. I want to find bounds on $E(XYZ)$. In the case of bounding $E(XY)$...
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1answer
54 views

Elliptic equation on square with periodic boundary values for the solution and it's partial derivatives

Suppose that $\Omega=[0,1]^2$. I will say that a real valued function $u$ on $\Omega$ satisfies periodic boundary values if $$u(x,0)=u(x,1), \;u(0,y)=u(1,y),\;\;\;\text{ for all }x,y\in[0,1].$$ Now ...
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2answers
122 views

Example of a linear operator whose graph is not closed

I want an example of a linear operator $T:X\to Y$ such that graph of $T$ is not closed. My thoughts: $T$ must be unbounded. Again by closed graph theorem any unbounded linear map from a Banach space $...
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1answer
300 views

Bounding the decrease after applying a contraction operator $n$ vs $n+1$ times

Can we upper bound the convergence rate of $$\max_{\textbf{v}: \left\Vert \textbf{v}\right\Vert_2=1} \left\{ \left\Vert \textbf{T}^n \textbf{v}\right\Vert^2_2 - \left\Vert \textbf{T}^{n+1} \textbf{v}\...
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69 views

Generator problem for reduced group C*-algebra

(Not sure if it is appropriate or not, if no I will delete the post) Recently I am concerned about the number of generator of $C^{*}_{r}(\mathbb{F}_{k})$, the reduced group algebra of the free group, ...
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3answers
537 views

Takesaki theorem 2.6

I originally posted this question on MSE and didn't get a satisfactory answer, even after putting a bounty on it. Hence, I thought I should ask here: Consider the following theorem in Takesaki's book &...
6
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1answer
162 views

Infinite-dimensional projections of linearly independent sets

A subset of a linear space $X$ is called infinite-dimensional if it is not contained in a finite-dimensional linear subspace of $X$. Problem. Let $L$ be an infinite-dimensional subset of the linear ...
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398 views

Why is spectral theory developed for $\mathbb C$

Spectral theory is a fundamental part of operator theory and the spectrum of many operators is investigated throughout the existing literature. And that is for a good reason: If $A$ is some closed ...
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1answer
72 views

Fundamental of a signal

Consider the space $S$ of real functions with the norm $$\|f\|^2 = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} e^{-x^2/2} f^2(x) ~\mathrm{d}x, $$ or any reasonable Euclidean norm such that bounded ...
8
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1answer
247 views

General validity of separation of variables

Let $L$ be any differential operator (not necessarily linear). Given initial conditions and boundary conditions (of any type), I am interested in general statements of the form: Given a boundary ...
5
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1answer
253 views

Making sense of the formula $\operatorname{Det} (I+M )= e^{\operatorname{Tr} \ln (I+M)}$, especially in the infinite dimensional cases

$\DeclareMathOperator\Det{Det}\DeclareMathOperator\Tr{Tr}$In physics literature dealing with quantum field theory, the formula \begin{equation} \Det(I+M) = e^{\Tr \ln(I+M)} \end{equation} appears ...
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1answer
155 views

How to choose minimisers in a continuous way

Let $\langle X, X' \rangle$ be a dual pair equipped with the weak and weak* topologies. Let $C$ be a weak* compact subset of $X'$ with nonempty interior. For each $x \in X$, let $M(x)$ be the set of ...
2
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0answers
65 views

Reference for Chebyshev centers

Today, I came across the concept of Chebyshev center twice. In particular, it is the key tool in the very elegant paper "A fixed point theorem for $L^1$ spaces" by Bader, Gelander and Monod. ...
4
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1answer
172 views

On some convergence theorems by Felix E. Browder (1967)

I have been reading Felix E. Browder's Convergence Theorems for Sequence of Nonlinear Operators in Banach Space and I was hoping I could find answers to a couple of questions I have about the paper. ...
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2answers
52 views

Construct suitable cutoff function

Let $\bar x \in \mathbb R$. Is there a cut-off function such that $\phi_\epsilon \in C^\infty(\mathbb R)$, $0 \le \phi \le 1$, and $$\phi_\epsilon(x) = \begin{cases} 1 &\text{ if } |x - \bar x| \...
2
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1answer
115 views

Sobolev embedding into measurable functions

Consider the fractional Sobolev space $$ W^{k,2}(\mathbb R^n):=\big\{f \in \mathcal S'\,\big|\,(1+\|\xi\|)^k\hat f(\xi)\in L^2(\mathbb R^n)\big\} $$ for some $k\in\mathbb R$, and let $\mathcal M$ ...
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0answers
48 views

Relationship between point spectrum and generalized inverses(Green operators) of elliptic self-adjoint operators

I'm very confused about the relationship between point spectrum and generalized inverses(Green operators) of elliptic self-adjoint operators. I'm following proof of a theorem in a paper, it seems that ...
1
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1answer
135 views

Does the Implicit Function Theorem in Banach spaces holds if the differential is only one-to-one (not onto!)?

Is the Implicit Function Theorem in the following form correct: Let $V_1,V_2,W$ be Banach spaces, and $Ω⊂V_1×V_2$ an open subset containing $(x_0,y_0)$. Let consider a continuously differentiable map $...
2
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0answers
95 views

Comparing two quantities related to the norm of an inner derivation

Let $M$ be a von Neumann algebra sitting in $B(H)$. Let $U(M)$ denote the unitary group of $M$. Let $I(M):=\{\tau\in M\,|\,\tau=\tau^*=\tau^{-1}\}$ the set of involutions in $M$. Let $SAC(M):=\{h\in M\...
2
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1answer
106 views

Reverse martingale convergence theorem in Banach spaces

In section 1.5 of a course given by Gilles Pisier, the author is claiming that in the excerpt below $\operatorname E[\varphi_i\mid\mathcal A_{-n}]\to\operatorname E[\varphi_i\mid\mathcal A_{-\infty}]$ ...
3
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0answers
27 views

Schmidt ellipsoids to different orthonormal bases

Let $H$ be a separable, infinite dimensional Hilbert space. For an ONB $(e_n)_{n \in \mathbb{N}}$ of $H$ together with a series $(\alpha_n)_{n \in \mathbb{N}} \subset (0,\infty)$ such that $\sum\...
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1answer
51 views

Iterated integrations by parts using the fractional Laplacian

Let $u \in C^\infty_c(\mathbb{\Omega})$ and $\varphi$ be an eigenfunction of the fractional Laplacian $(-\Delta)^s$ in $\Omega$ with eigenvalue $\lambda$. In what sense, if any, is it true that $$\...
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1answer
228 views

Support of a measure

Let $T:X\to X$ be a continuous function on a compact manifold $X$ and let $\text{Leb}$ be the Lebesgue measure normalized so that $\text{Leb}(X)= 1$. We denote by $\mathcal{M}(X)$ the space of all ...
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0answers
50 views

Is it possible to define a Bochner integral for a $S'(\mathbb R^d)$-valued function?

I apologize in advance for the rather vague question. While reading the book White noise distribution theory by H.H. Kuo, in particular the section 13.3 I came across the following statement (I'll ...
3
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1answer
130 views

Strong convexity inequality w.r.t. infinity norm $\lVert\cdot\rVert_{\infty}$

Consider a twice differentiable 1-strongly convex function $f:\mathbb{R}^n \to \mathbb{R}$. Is it true that there exists $\alpha>0$ independent of $n$ such that, for all $x \in \mathbb{R}^n$: \...
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28 views

Conditions on $g$ which ensure the function $\phi(t):=\int_0^{2\pi} g(\cos\theta)g(\cos(\theta-\arccos(t)))\,\mathrm{d}\theta$ is $C^k$ on $(-1,1)$

Given an almost-everywhere continuous function $g:[-1,1] \to \mathbb R$, define another function $\phi_g:[-1,1] \to \mathbb R$ by $$ \phi_g(t) := \int_0^{2\pi} g(\cos\theta)g(\cos(\theta-\arccos(t)))\,...
2
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1answer
57 views

Elliptic regularity when the Lagrangian is possibly infinite

I want to solve variational problems of the form $$\inf_u \int_{-1}^1 \phi(u'(x)) \text{ with } u(-1)=u(1) = 0,$$ where $\phi(p)$ is convex and is allowed to take on the value $+\infty$ for some ...
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0answers
102 views

Is the kernel $\vert d_X - d_Y \vert^p$ conditionally negative definite?

Given two finite metric spaces $(X,d_X)$ and $(Y,d_Y)$, for $p > 0$, define the kernel ($4$-D tensor) $K$ on $(X \times Y)^2$ by: $$K\big( (x_i, y_k), (x_j, y_l) \big) = \vert d_X(x_i, x_j) - d_Y(...
4
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1answer
153 views

Functions in $H^1(\mathbb{R}^2)$ but not in $C(\mathbb{R}^2)$? [closed]

I'm struggling to come up with examples of functions which are in $H^1(\mathbb{R}^2)$ but not in $C(\mathbb{R}^2)$. I know that $H^1(\mathbb{R})\subset C(\mathbb{R})$, and I know the power law example ...
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1answer
53 views

On level sets of smooth functions in a bounded domain

Let $\Omega$ be a bounded domain in $\mathbb R^n$, $n\geq 2$, with a smooth boundary and let $f$ be a smooth function on $\bar\Omega$. Is there a natural condition that one can impose on $f$ ( say in ...
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45 views

A question about parametrix

$\DeclareMathOperator\id{id}$Let $D$ be a differential operator, and $Q$ a parametrix of $D$, i.e., $$ QD=\id-S_0 ,\qquad DQ=\id-S_1$$ where $S_0$, $S_1$ are operators with smooth kernels. A special ...
0
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1answer
96 views

Continuous surjection of $\mathbb{R}^{n-1}$ onto the interior of the $n$-simplex with continuous right inverse

Let $n$ be a positive integer. Clearly $\mathbb{R}^{n-1}$ and the interior of the $n$-simplex $ \delta_n := \{x \in [0,1]^n:\,\Sigma_k x_n =1, (\forall i)\,x_i>0\} $ are homeomorphic. What I'm ...
2
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1answer
83 views

Change of variable and boundary data for Laplace equation

Let us consider a smooth bounded domain $\Omega \subset \mathbb R^n$ and the problem $$ \begin{cases} -\Delta u = 0 & x \in \Omega \\ u = 1 & x \in \partial \Omega \end{cases} $$ Does it make ...
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0answers
34 views

Can RKHS of Gaussian kernels over $\mathbb R^d$ have a non-zero element which is zero on a subspace $\mathbb {R^{\mathit k}\subset R}^d$ where $k>0$?

I have originally asked this question on math.sx but thought maybe here is actually a better place to ask it. Please do let me know if it is actually off-topic for mathoverflow. I initially thought ...
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0answers
67 views

Logarithmic Sobolev inequality for a probability measure

Is there a log-Sobolev inequality for a probability measure without constraints on the function? (Managed to find for Lipschitz functions but not for any differentiable function and any probability ...
3
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0answers
128 views

de Rham currents/distributions on manifolds with boundaries

My main source for currents and distribution theory on manifolds in general is de Rham's Differentiable Manifolds. To recap, let $M$ be a smooth, $m$ dimensional real manifold without boundary. De ...
2
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1answer
154 views

Viscosity solutions of $(-\Delta)^s u = 0$ in $\Omega $ with non-homogeneous data $u = 1$ in $\mathbb R^n \setminus \Omega$

Let us consider a smooth bounded domain $\Omega \subset \mathbb R^n$ and the problem $$ (1) \quad \begin{cases} (-\Delta)^s u +\lambda u= 0 & x \in \Omega \\ u = 1 & x \in \mathbb R^n \...
5
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1answer
128 views

A perturbation of an unconditionally convergent series in $\ell_2$

For two functions $x,y:\omega\to\mathbb R$ let $xy:\omega\to\mathbb R$, $xy:n\mapsto x(n)y(n)$, be their pointwise product. It is clear that for any elements $x,y\in\ell_2$ their pointwise product $xy$...
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0answers
21 views

Interpolation spaces defined by singular value decomposition

Let $ X $ and $Y$ be Hilbert space, $A:X \to Y $ compact and injective, $(\sigma_n;v_n,u_n)$ be its singular value decomposition, that is, $$ Av_n = \sigma_n u_n \\ A^* u_n = \sigma_n v_n $$ Since $\...
0
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0answers
25 views

Is these some transformation that transforms an arbitrary $\mathcal C^1$ function into an $L$-smooth function?

Let $\mathcal X\subset \mathbb R^d$. We say a function $f:\mathcal X\to \mathbb R$ is $L$-smooth if it is continuously differentiable and $\|\nabla f(x)-\nabla f(y)\|\le \|x-y\|$ holds for all $x,y\in ...
5
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1answer
278 views

Definition of infinite-dimensional Gaussian random variable

For infinite-dimensional Gaussian measures, we often see the definition of Gaussian random variables like this: Let $H(\Omega;\mathbb{R})$ be a separable Hilbert space. A random variable $u \in H$ is ...