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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23 views

Inequality in geodesic quadrupel

Let $(M,g)$ be a compact complete Riemannian manifold. Consider the geodesic quadrupel $ABCD$ where $l:=d(A,B)=d(B,C)=d(C,D)=d(A,D) \geq \frac{1}{2}$ and $d(B,D),d(A,C) \geq 1$. Let $x \in CD$ and $y ...
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75 views

How to represent $\limsup \cdot \liminf$ of Booleans

Let $X$ be some set, and let $A,B\subset X$. By $1_A(x)$ let us mean the indicator/characteristic function. Let $(x_n)_{n\in \Bbb N}$ be some sequence in $X$. I have an expression of the form $$ ...
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44 views

Example of weakly discontinuous contraction

Can somebody give an example of a projector $P_c$ on the convex closed set C which is not weakly-continuous?
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56 views

Prove operator $T$ is onto

Consider the Hilbert spaces $X := H^{1}(\Omega)\times H^{1}(\Omega)$ and $Y:=L^2(\Omega)\times L^2(\Omega)$, where $\Omega =\ ]{-}\pi, \pi[$, and \begin{eqnarray*} \langle(u,v), (z,w)\rangle_X & = ...
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51 views

Can you construct a coutable local base in the space of continuous functions?

Let $(C,\tau)$ be the topological vector space of all complex continuous functions on $[0,1]$ with seminorms $p_x(f)=|f(x)|$, $x\in [0,1]$. We have known $(C,\tau)$ is not metrizable,but how could I ...
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52 views

Does a limit exist in a non-metrizable space?

Let $C$ be the vector space of all complex continuous functions on $[0,1]$. And let $(C,\tau)$ be the topological vector space defined by the seminorms $$p_{x}(f)=|f(x)|~~~~(0\leq x\leq 1).$$ We have ...
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161 views

Compact integral operator

I have this exercise and I don't know how to solve the last question. In the following $a,b$ are two real numbers such that $a<b$ ,$E=C([a,b],\mathbb{R})$ with the norm $||.||_0$ given by ...
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35 views

convergent sequence in quadratic mean and distrubutional sense?

"Say if the following sequence of functions in R: f_n(x) = { 0, if |x| < n; exp(−|x|/n), if |x| > n. converges (1) in quadratic mean, (2) in the sense of distributions." My own calculation of (1) ...
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43 views

Eigenvalue of Linear differential operator with boundary conditions

I am wondering if there is a way to find the eigenvalues for some simple-looking equations without solving the equations explicitly. More specifically, the equation(s) are as following: ...
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78 views

Contraction Mappings

I'm self-learning functional analysis at the moment and although I can understand the underlying theory I have difficulty applying it in the aggregate. Can someone please break this down for me step ...
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52 views

Total set of functions in $L^2(\Omega)$

Are the sets of functions $\{e^{\int_0^T h(s)dB_s -\frac{1}{2}\int_0^T h(s)^2 ds}\}$ and $\{e^{\int_0^T h(s)dB_s}\}$ total in $L^2(\Omega)$? What is the difference? What should one use to prove weak ...
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34 views

$\langle f, v \rangle_{L^2(0,T;H'), L^2(0,T;H)}=0$ for all $v$ implies $f = 0$?

Suppose that for some $f \in L^2(0,T;H')$, $$\langle f, v \rangle_{L^2(0,T;H'), L^2(0,T;H)}=0$$ for all $v \in L^2(0,T;H).$ How do I show that this implies $f = 0$? $H$ is Hilbert.
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71 views

Is this argument correct to show that I have an isometric isomorphism?

I have a map $T:X \to Y$ between Hilbert spaces. I want to show that $T$ is isometrically isomorphic. I have found: 1) For every $x \in X$, we have $Tx \in Y.$ 2) For every $y \in Y$, there is an ...
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52 views

Simple representation of measurable maps w.r.t. different measures

Let $(\Omega, \mathcal F)$ be a measurable space and $f:\Omega\rightarrow\mathbb R$ a bounded and measurable function. Let $\mathbb P_1,\dots,\mathbb P_N$ be any probability measures. It is well ...
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42 views

variational formulation

I have this exercise. We put $X = L^2(\Omega)^2 \times L^1_0(\Omega), M = H^1_0(\Omega)^2.$ $$a : X \times X \rightarrow \mathbb{R}; ((\sigma,p),(\tau,\alpha)) \rightarrow (\sigma \otimes \tau)$$ ...
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58 views

Arveson index of a completely positive map on matrix algebra

Can someone tell me what is Arveson index of a completely positive map. What I want is given a map \begin{eqnarray} \psi:\mathcal{B}(\mathbb{C}^m)&\longrightarrow&\mathcal{B}(\mathbb{C}^n)\\ ...
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47 views

$\sum_{m=-l, …,l; l=0,1,2,..} e^{\frac{-i E_l (t_f-t_i)}{\hbar}} Y_{lm}(\phi_f,\theta_f)Y_{lm}(\phi_i, \theta_i)$

I have encountered this series while trying to calculate the path integral of a free particle on a sphere. The sum is $$K=\sum e^{\frac{-i E_l (t_f-t_i)}{\hbar}} Y_{lm}(\phi_f,\theta_f)Y_{lm}(\phi_i, ...
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99 views

Question about space of sequences.

Consider the complex linear spaces $l_{1}, l_{\infty }$ and the subspace $c_{0}$ of $l_{\infty }$ sequences consisting of $\left ( x_{n} \right )_{n\in \mathbb{N}}$ such that $\lim_{x \to 0}=0.$ Show ...
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56 views

$\ast$-homomorphism

Let $\phi: C(X,M_{4}(\mathbb{C})) \rightarrow C(Y,M_{8}(\mathbb{C})) $ be a $\ast$-homomorphism where $X$ and $Y$ are compact Hausdorff spaces. Let $M_{2}(\mathbb{C})$ be the C*-subalgebra of ...
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32 views

Continuity of a certain matrix-like function

Let $X$ be a finite set and let $M$ be a space of all probability measures on $X$. Let $f:X\to\Bbb R^{m\times m}$ be a random matrix and consider a function $c:M\to\Bbb R$ defined as $$ c(\mu) := ...
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51 views

Hahn Banach to get linear functional bounded by sub/superlinear functionals

I am working in a real vector space $V$. I have seen it written that if I have a sublinear functional $p$ and a superlinear functional $q$ such that $q \le p$ then there exists some linear functional ...
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114 views

Does every Banach space admit a continous injection to a non-closed subspace of another Banach space?

Let $(V, ||\,||)$ be a Banach space. I want to produce a non-complete norm $||\,||'$ on it such that $||v||' \leq ||v||$ for all $v$ in $V$. Given a continuous injection $\varphi\colon V \to W$ with a ...
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43 views

Is the Cauchy principal value “invariant” under change of variables?

Let $f \in C^{\gamma}_c(\mathbb{R}) $. Let $K:\mathbb{R}^n \backslash \{\vec{0}\} \rightarrow \mathbb{R}^n$ be a singular integral kernel with the following properties: 1) K smooth everywhere except ...
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1k views

Finding gradient of a norm

How to calculate the gradient of a function $f(\mathbb x) = \| \mathbb x \|$ where $\mathbb x$ is a $n$ dimensional vector, $\|\cdot\|$ could be either a $L_1$ norm or a $L_2$ norm or a ...
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16 views

normal weights acting on separable Hilbert spaces.

a normal weight is often definied as a sum of normal positive functionals. question: If we have a normal weight, definied as above, on a von neumann algebra acting on a separable hibert space can ...
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65 views

Haar system and martingale

Today our lecturer used martingale theory to show that the Haar system is a basis for $L_1[0,1]$ (we're operating on the probability space $([0,1],\mathcal{B},P)$, where $P$ is the Lebesgue measure). ...
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95 views

Why is a *-homomorphism isometric, if it maps strictly positive elements to strictly positive elements?

I have the following exercise: Let $\pi:\mathcal A \rightarrow \mathcal B$ be a *-homomorphism between two unital $C^*$ algebras $\mathcal A$ and $\mathcal B$ which maps the unit to the unit. Assume ...
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138 views

Regularization by mollifier sequences

A well-known feature used in PDE's is the regularization by convolution with a mollifier sequence $\rho_n$, i.e. $\rho_n(x) := n^d \rho(nx)$ with $x \in \mathbb R^d$, $\rho \in C^\infty_c(\mathbb ...
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31 views

The tightest bound on an integral

Consider a polynomial $p(x)$ where $p(x)>0$ for $x\in(0,1)$ and $p(0)=0$. Let $s(x)$ be an increasing analytic function such that $s(0)=0$ and $s(1)=1$. I am interested to bound the following ...
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94 views

adjoint operator in Sobolev space

Let $g\in H_0^1(\Omega)\cap W^{2,\infty}(\Omega),$ and let us define the operator $B : y \to g y$ from $H:=H_0^1(\Omega)\cap H^2(\Omega)$ to $H$, which we endowed with inner product : $<u,v>_H ...
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56 views

For each Hermitian operator $H $ the operator $U:=(H-iI)(H+iI)^{-1}$ is unitary.

For each Hermitian operator $H $ the operator $U:=(H-iI)(H+iI)^{-1}$ is unitary. I've come up to this and don't know how to proceed. $U^{H} = (H-iI)^{-1}(H+iI)$ ...
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40 views

Compactness in $L^p$

I am studying this article: http://arxiv.org/pdf/0906.4883.pdf There is a little part that I do not understand, in the proof of theorem 5, page 4. Let P be the projection map of $L^p(\mathbb{R}^n)$ ...
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86 views

Find spectrum of the operator and an explicit form for the solution

Let $A: L^2[0,2\pi] \to L^2[0,2\pi]$ be defined by $Au=v$ if $v(x)=\int^{2\pi}_{0} cos(x+t)u(t)dt$ (a) Find the point spectrum, eigenspace, residual spectrum, continuous spectrum, and spectrum (b) ...
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146 views

Find spectrum of the operator

I need some help to find point spectrum, residual spectrum, continuous spectrum and for this problem. Let $\theta \in\mathbb{C}$. On the complex space $L^2[-1,1]$ consider the operator $Au=v,$ ...
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131 views

Properties of the derivative $\frac{\delta F(\phi(x))}{\delta\phi(y)}$ , where $ \phi(x) = F(\phi(x)) $ is a fixed point problem.

Dear experts I have a fixed point problem of the type: $ \phi(x) = F(\phi(x)) $, where $x \in \mathbb R^3 $. $\phi(x)$ is differentiable non-negative function on a given domain. I am trying to find ...
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241 views

Are continuous functions strongly measurable?

Measure theory is still quite new to me, and I'm a bit confused about the following. Suppose we have a continuous function $f: I \rightarrow X$, where $I \subset \mathbb{R}$ is a closed interval and ...
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111 views

best approximation property of finite dimensional banach space.

recently i am stuck in a question whose partly answer is known to me. But i want full answer. Can anyone help me? The question is following: Suppose $X$ is a finite dimensional normed linear space ...
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34 views

“Compensated weak compactness” in $W^{1,1}$

In a problem I'm working on I want to show that a bounded sequence $\{u_n\} \subset W_0^{1,1}((0,1))$ converges weakly. Of course, since $W^{1,1}$ is not reflexive I don't get this for free. The ...
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36 views

Weak limits and structure of a generated semigroup

I am getting acquainted with the beautiful theorem known as Jacobs–de Leeuw–Glicksberg decomposition. A special case of this theorem is the following: Theorem. (Jacobs–Glicksberg–de Leeuw ...
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160 views

Application of Stone Weierstrass Theorem for trigonometric polynomials

In the space $C[-\pi, \pi]$ equipped with the sup norm, consider the linear space $M$ spanned by the functions ${ (\cos nx)}_{n\geq0}$ and $ {(\sin nx)}_{n\geq1}$. what is the closure $\bar{M}$ of ...
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56 views

How to prove: Every non-degenerate representation $\{ \pi , H \} $ of an involutive Banach algebra is a direct sum of cyclic representations.

I am having trouble with the following proof: Prove: Every non-degenerate representation $\{ \pi , H \} $ of an involutive Banach algebra is a direct sum of cyclic representations. ........... ...
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32 views

System with bounded delays : uniqueness theorem.

Let $F:[t_0,\beta) \times C_D \rightarrow\mathbb{R}^n$ continuous and Locally Lipschitzian. then, given any $\phi \in C_D$, eqs. $x'(t)=F(t,x_t)$ and $ x_{t_0}=\phi $ (initial value condition) have at ...
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124 views

How can projection operators be limits of powers of unitary operators?

Consider a (fixed) unitary operator $U$ acting on the Hilbert space $\mathcal{H}$. Because the unit ball is compact in the weak topology, it is not hard to see that there exists a (smallest) compact ...
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47 views

All finite Baire measures are Closed-regular?

Given a finite Baire measure $\mu$ on a topological space $X$, is it true that $\mu$ is closed-regular? Where closed regular means that, $$\mu(A) = \sup\{\mu(K)| K \space \text{is a } Z\text{-set}, ...
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61 views

Verify solution: Is this gradient, correct?

For a function $$f(X)=\operatorname{tr}(X^TAX)+\|\operatorname{diag}(X^TX)-\alpha I\|_2,$$ where all entries are real and $\alpha$ is a real scalar, while $A$ is a p.s.d matrix and $X$ is a real ...
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123 views

From positive definite function to Følner sequence -— a question on amenability and nuclearity

We know that amenability of countable discrete group $\Gamma$ has many equivalent characterizations. In particular, there are two: a) there is a sequence of finitely supported positive definite ...
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45 views

What is the matrix norm in defining the generator of a continuous time Markov chain?

For a continuous time Markov chain with finite state space and Markov transition function $p(t)$, its generator $G$ can be defined entry-wise as $$ G_{i,j}:=\lim_{t\to 0^+} \frac{p_{i,j}(t) - ...
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70 views

Is $C_0^\infty(0,T;H)$ dense in $L^2(0,T;H)$?

Let $H$ be a Hilbert space. Is the space $\mathcal{D}(0,T;H) = C_0^\infty(0,T;H)$ dense in $L^2(0,T;H)?$ I am aware that this space is dense in the space of differentiable Bochner functions but I ...
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95 views

Harnack's Inequality and (hypo)elliptic PDE

Background: I am aware of the Harnack's Inequality for linear elliptic equations. My questions are: (a) Is there a version of Harnack's Inequality for nonlinear elliptic equations, say, of the form ...
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67 views

How to deduce the results of response time by this trajectory approach?

First, we denote this: And And we get this right property( $last_i$ means the last node on $τ_i$): And: $Smin_i^h$ = $\sum_{h'=first_i}^{h-1} ({C_i^{h'} + L_{max})}$ $Smax_i^h$ = ...