Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, as well as measure, ...

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Equation Involving Bilateral Laplace Transform

Assume that $f(x,y)$ is a compactly supported, joint probability density function on $\mathbb{R}^2$ and nice enough for the following function to make sense: $$P_t(y):=e^{ty}-\int_{-\infty}^\infty ...
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69 views

Continuity of a induced representation of $C_c^\infty(G)$

Let $G$ be a Lie group and $(\pi,V)$ be a continuous representation of $G$ on $V$ a Fréchet space. Let $dx$ denote the Haar measure on $G$. The representation $\pi$ induces a representation of ...
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126 views

A question using Baire Category Theorem

If $f\in C^\infty(\mathbb{R},\mathbb{R})$ is a smooth function and assume that $f$ restricted to $[a;b]$ is not a polynomial for all intervals $[a; b]\subset \mathbb{R}$ with $a < b$. Prove that ...
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97 views

Compactness of the set of densities of equivalent martingale measures

Consider an incomplete market $(\Omega,\mathcal F,\mathbb P)$ driven by a semimartingale $S=(S_t)_{t\in[0,T]}$. Under the no free lunch under vanishing risk (NFLVR) assumption, the set $\mathcal ...
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153 views

The separation theorem for the w* topology

When I read the prove of the Goldstine Theorem(See An Introduction to Banach Space Theory Robert E. Megginson 2.6.26), I find it use the separation theorem for the w* topology without any details. ...
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237 views

Orthogonal projection and normal operators

Let $G$ be normal operator with compact resolvent such that $\ker G$ is different from $\{0\}$. Now Let $P$ be the orthogonal projection onto $\ker G$ and consider $G' = G + P$. Please, I want an ...
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122 views

Uniform Integrability on Compact sets

Let $m$ be a probability measure on the compact set $W \subset \mathbb{R}^m$, so that $m(W)=1$. Consider $f: X \times W \rightarrow \mathbb{R}_{\geq 0}$ locally bounded, $X \subseteq \mathbb{R}^n$, ...
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69 views

Question (3) on Uniform Integrability (simpler)

Let $m$ be a probability measure on $W \subseteq \mathbb{R}^m$, so that $m(W)=1$. Consider $f: X \times W \rightarrow \mathbb{R}_{\geq 0}$, $X \subseteq \mathbb{R}^n$ such that $\forall w \in W$ $\ ...
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233 views

Question (2) on Uniform Integrability

Let $m$ be a probability measure on $W \subseteq \mathbb{R}^m$, so that $m(W) = 1$. Consider $f: X \times W \rightarrow \mathbb{R}_{\geq 0}$ locally bounded, $X \subseteq \mathbb{R}^n$, such that ...
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69 views

Is there chance to form a frame (Riesz basis)?

Let $f$ be a continuous function on $\mathbb{R}$ with compact support with exactly one maximum. Form the functions $$ f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right) $$ One can show that ...
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144 views

Is $C_0^\infty $ dense in $W^{m,p}$?

Is $C_0^\infty $ dense in $W^{m,p}$? Here $C_0^\infty$ = $C_c ^ \infty$ : $C^\infty$ with compact supports, and $W^{m,p}$ : Sobolev spaces.
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69 views

Dense property of $C^k_0(\Omega)$

When $\Omega = \mathbb{R}^d$ or $\mathbb{R}^d_+$, $C^k_0(\bar{\Omega})$ is dense in $W^{k,p}(\Omega)$(Sobolev space, k-differentiable and each term estimated by p-norm). I am wondering if it holds ...
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83 views

Least fixpoint in a Banach space of bounded measurable functions

Let $(E,\mathscr E)$ be a measurable space and denote by $\mathrm b\mathscr E$ the space of all Borel measurable bounded functions $f:E\to\mathbb R$. On this space the partial order is given by $$ ...
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167 views

A question about weak lower semicontinuity

Let $\Omega \subset \mathbb{R^{n}}$ be a bounded domain and $u , u_j \in H^{1}(\Omega)$ such that $u_j \rightharpoonup u$ in $H^{1}(\Omega)$ $$ F_{1}(u) = \int_{\{ u > 0\}} \dfrac{1}{2}\langle A_1 ...
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66 views

Functional Analysis- Convergence

Given an operator $H$ , and a sequence $\{ H_n \} _{n\geq 1 } $ in an arbitrary Hilbert Space , such that both $H$ and $ H_n $ are self-adjoint and non-negative. How can I prove that ...
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71 views

How to define operators on $\ell^p_{00}$?

Given $p\in (1,\infty)$. Take a bounded sequence $(f_n)$ in $\ell^p$ and define a linear map $T\colon \ell^p_{00}\to \ell^p$ ($\ell^p_{00}$ is the space of finitelty supported sequences in $\ell^p$) ...
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116 views

An integral transform.

Let's consider a complex function that can be represented in the following form: $$ K(z)=\int_{-\infty}^{\infty}A(\alpha)z^\alpha d\alpha $$ Writing $z=re^{i\theta}$, we get: $$ ...
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107 views

Partition of unity. Functional analysis

Need to find partition of unity in case of oparator $A_{f}(x)=(|x-1|+x)f(x)$. Operator $A \in L_{2}[0,2]$ Partition of Unity is set of operators $E_{\lambda}=E((- \infty,\lambda]) $, where ...
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101 views

Graph measurability (difficult but interesting)

Let $\mu(\cdot)$ be a probability measure on $W \subseteq \mathbb{R}^p$, so that $\int_W \mu(dw) = 1$. Consider a locally bounded function $f: \mathbb{R}^n \times \mathbb{R}^m \times W \rightarrow ...
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203 views

Application of method of continuity in partial differential equations

Consider a differential operator $$L_t:= (1-t)(\Delta-\lambda) + t L,\qquad t\in[0,1].$$ For any $u\in C^2_0(\mathbb{R}^2)$, we have $$\lambda^2 \|u\|_2^2 + 2\lambda\sum_{i}\|u_i\|_2^2 + ...
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112 views

Induction on Uniform Boundedness

This question gives interesting insights on whenever uniform boundedness can be "iterated". Let $\mu(\cdot)$ be a probability measure on the closed set $Z \subseteq \mathbb{R}^p$, so that $\int_Z ...
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217 views

Strongly exposed points/Exposed points

I was studying and I got the next doubt: We suppose that $(X,\|\cdot\|)$ is a Banach space and $C$ it is a convex closed subset of X. We say that $x\in C$ it is an exposed point of $C$ if $\exists ...
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78 views

tensor product, notation, and the sup norm

This question is a bit tricky for me to post because I don't really understand all the symbols and techniques involved - so it is likely that I'll do mistakes. In case it is unclear what I write ...
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159 views

Convolution of measures and Fourier transform of a finite measure

I am reading a book on Harmonic Analysis on $\mathbb R^n$. It need some facts about finite measure space $\mathcal B(\mathbb R^n)$, which is said to be the dual of $C_0(\mathbb R^n)$. In this space we ...
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86 views

positively homogeneous asymptotic expansion associated to the symbol of a pseudodifferential operator

I am currently reading about pseudodifferential operators and their symbols, and I came across the notion of classical pseudodifferential operators. For these it is possible to find an asymptotic ...
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371 views

Haar-measure on the torus

Good evening! Let $ \mathbb{T}:=\{ z \in \mathbb{C} ; \vert z \vert =1 \} $ be the unit circle in the complex plane. We denote the trace Borel-$\sigma$-algebra on $\mathbb{T}$ by ...
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91 views

Markov-like inequality for functionals

Dear fellow mathematicians, The Markov inequality reads, for $(\Omega, \mathcal{F}, \mu)$ being a measure space, and $f$ a real valued function on $\Omega$ (you can also see Stein, Singular Integrals ...
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108 views

Easy way to express $\sin(x)$ by basis $\sin(a\pi x)$

I am again working through some old assignments for which I unfortunately have to solutions. The assignment ist: Find the formal solution on $\Omega = ]0;1[^2$ for $-2 \frac{\partial^2u}{\partial ...
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62 views

Represent the generalised function by the function

Consider a function $a(x,y,\xi) \in C^{\infty}(\mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}^n)$ such that $a(x,y,\xi) \leqslant C(1 + |\xi|)^m$ and its derivatives satisfy the same inequality ...
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64 views

$C^{\infty}$ function represented by the diverging integral

There is a theorem (see Treves: "Introduction to Pseudodifferential and Fourier integral operators") that states that the kernel of any pseudodifferential operator, i.e. the distribution $$ K(x,y) ...
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82 views

Extension of closed linear functionals…

If $X$ is a normed linear space and $X^*$ be its completion, consider a linear functional $f$ belonging to $X'$ which is a closed map. By Hahn-Banach extension there exists a linear functional ...
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69 views

Continuity of a pointwise greater-than-zero mapping

I'd like to ascertain the continuity of a mapping with respect to a parameter. Consider the function $$ g(u(x)+\gamma v(x)) = \left\{ \begin{array}{l} 1 & u(x)+\gamma v(x) > 0 \\ 0 & ...
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64 views

Show properties of elements of $\mathcal{H}^2$

I have problems with my homework. First of all: $\mathcal{H}^2$ means the Sobolev space, right? Because in the script we are using a $W$ for Sobolev spaces and the $\mathcal{H}^2$ can't be found ...
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180 views

Sum of inner products in a hilbert space

This question asks to show that the following inner product defines a Hilbert space/is complete: $$\langle f,g\rangle=\sum\int_0^1 f^{(k)}\mathrm{conj}(g^{(k)})$$ where $f^{(k)}(t)$ are continuous on ...
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193 views

Criterion for a countable family of functions to separate points

So this may be a weird/dumb question, but I was wondering whether a sufficient condition is known for a (countable) family of continuous functions to separate points on, say, a compact metric space ...
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102 views

Existence of 'Lower bound' functions for convex functions on radially open convex sets

I am trying to solve a problem which is as follows: Suppose $f$ is a convex function on a radially open convex subset $C$ of a vector space $E$, and $x \in C$. Show that there exists a linear ...
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153 views

Neumann series in an incomplete normed algebra

Let $\mathcal{A} \equiv (A, \|\cdot\|_A)$ be a unital (associative) normed algebra over the real or complex field, and assume that $\mathcal{A}$ is not complete. Provided $\mathcal{B}_\mathcal{A}$ is ...
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337 views

Weak derivative

Let $u \in C(\Omega)$ be a function with weak derivative $Du \in C(\Omega)^n$. How does one prove that $Du$ coincides with the classical derivative? Is the mean value theorem for integration ...
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109 views

How can I compare unbounded linear operators?

Let $X$, $Y$ be Hilbert spaces. Let $S, T : X \rightarrow Y$ be unbounded operator. Suppose $S$ and $T$ be bounded operators. Then we can compare by their maximum distance on the unit ball of $X$. ...
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242 views

Explanation for something in a Functional Analysis book by Kesavan

My wife is reading "Functional Analysis and Applications" by Kesavan. On page $140$, at the bottom it says "Let $\Omega$ be an open set, $f\in L^2(\Omega)$, and $u\in H^1_0(\Omega)$ such that ...
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634 views

cokernel and kernel of adjoint operator

Let $L$ be a linear operator from Banach space $X$ to $Y$. Is the dimension of the kernel of the adjoint of $L$ the same as the dimension of the cokernel? The cokernel is $Y/(Im L)$. Also, is the ...
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75 views

Fourier series of $f(x,y)$

I want to find out whether the Fourier series of $\partial_x^i f$ and $\partial_y^i f$ converges absolutely if $f$ is a function in $L^2$ and both of its fourth partial derivatives exist and are ...
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193 views

Is the operator Fredholm?

Let $A$ be an operator on a Banach space, possibly unbounded, such that its resolvent $(\lambda - A)^{-1}$ is compact. Is $A$ then a Fredholm operator of index 0? My feeling is yes but I cannot prove ...
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134 views

solution for related function

If we have $f(x)=(2A(A+\delta)-1)x^{2}-2Ax+1$, then the values of $x$ are equal to $\frac{1}{A+\sqrt{1-A(2\delta+A)}}$ and $\frac{1}{A-\sqrt{1-A(2\delta+A)}}$. The question is how to find the ...
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135 views

Countable product of Hilbert spaces

Let $H_1, H_2, \ldots $ be a countable set of Hilbert spaces. Let $H\subset \prod_k H_k$ be the set where $$\|x\|^2 = \sum_k \|x_k\|^2_{H_k} < \infty.$$ Show that $H$ is a Hilbert space. It ...
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95 views

Are nuclear Montel spaces projective?

The well known fact about $\ell^1$ says that the Schauder basis of $\ell^1(I)$ behaves more-less like a Hamel basis, namely if $X$ is any Banach space and $\mathcal{E}=(e_i)_{i\in I}$ is a basis for ...
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99 views

Does a projection valued measure (PVM) induce a PVM on a generic subspace of the Hilbert space?

Let $E:{\cal B}(X) \to Pr({\cal H})$ be a projection valued measure (PVM), where ${\cal B}(X)$ is the Borel $\sigma$-algebra of a suitable topological space $X$ and $Pr({\cal H})$ is the set of ...
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134 views

One special set on [0,1]

Let $X = [0,1]$. Define $f:X\to\mathbb{R}_{\geq 0}$ to be Lipschitz continuous on $X$. Put $$Y\subset X:\int\limits_Y f(x)\,dx = 0$$ What can we say then about $A = X\setminus Y$? It is not defined ...
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109 views

reference search

Hello everyone I am looking for a couple of references: Claim 1 states that an open and connected set in $R^n$ is path-connected. Or more general an open, connected and locally connected set is ...
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91 views

Variability of a curve with Vapnik-Chervonenkis dimension 4

Say I have 10,000 data in 2-D and I want to fit a curve to them. There are many functional forms this curve could take -- polynomial, B-spline, trigonometric, and so on. I've decided that I only want ...