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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A convergence in norm topology

Let $H$ be a Hilbert space and $P_{n}\in B(H)$ be an increasing net of finite-rank projection which converge to the identity in the strong operator topology. Then, for any $T_{1}, T_{2}\in B(H)$, ...
3
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170 views

Best approximation for a normed vector space $X$

I am self-studying functional analysis. As far as I know, a best approximation of $X$ by a closed subspace $C \subseteq X$ exists and is unique if $X$ is a Hilbert space, a uniformly convex Banach ...
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2answers
72 views

Distance between two sets

Let $A,B$ be nonempty sets . $ D(A,B)=\inf\{D(a,b) : a\in A , b \in B \} $, let $C=\operatorname{cl}(A) E=\operatorname{cl}(B)$ now how can I prove that : $D(A,B) = D(C,E) $
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147 views

Contour integration of Resolvent

Let $B$ be a Banach space and $Q\colon B\to B$ be a linear operator with eigenpairs $\{(q_{k}, v_{k})\}$ with $v_{k}$ orthonormal. In this document, it is shown that if $C$ is a contour containing ...
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137 views

Determine if these two norms are equivalent

Let we have the space $C[a,b]$ (the space of all functions that are continuous on closed interval $[a,b]$). And we have two norms on this space: $$\|X\|_1= \max_{t\in [a,b]} | x(t) |$$ $$\|X\|_2 = ...
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63 views

Extensions in parabolic Hölder spaces

Let $\alpha\in ]0,1[,k\in\mathbb{N}.$ Let $\Omega$ be a open and bounded subset or $\mathbb{R}^n$ of class $C^{k+\alpha}$. As one could find in G.M. Troianello "Elliptic Differential Equations and ...
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2answers
66 views

Obtain solution of boundary problem as linear operator.

I'm kinda stuck with a problem right now. I have the boundary problem $$\left\{ \begin{array}{l} -u''(x)+\mu u(x)= f(x), \quad x\in (0,T) \\ u'(0)=u'(T)=0 \end{array} \right.$$ and I have to obtain ...
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34 views

Is the $\sigma$-finiteness condition necessary to ensure that $L^p(\mu)$ is reflexive?

Suppose $(X,\mu)$ is a measure space, and $p,q>0$ such that $1/p+1/q=1$. We know that if $\mu$ is $\sigma$-finite, then $L^p(\mu)$, $L^q(\mu)$ are reflexive and dual to each other. The proof could ...
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44 views

Existence Adjoint Operator: Boundedness?

Context This would make the check on the GNS construction much more simple. Problem Given a Hilbert space $\mathcal{H}$. Consider a merely linear operator $A:\mathcal{H}\to\mathcal{H}$. Suppose ...
6
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101 views

$\left\{\,f\in L^1[0,1]\,\big\vert\,\int_0^1\lvert f\rvert^2>1\right\}$ is open

Problem (Here $\lVert f\rVert$ means the $L^1$-norm $\int_0^1\lvert f\rvert$) Suppose $f\in L^1[0,1]$ such that $\int_0^1\lvert f\rvert^2>1$. I need to work out an explicit $\epsilon>0$ such ...
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2answers
77 views

Doubt on Arzela-Ascoli theorem

Consider a sequence of equicontinuous and uniformly bounded functions on a compact set. Under which condition I can say that it has a unique uniformly convergent subsequence ? Or, atleast uniform ...
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1answer
41 views

For $u\in H^s(\mathbb{R})$ with $s>n/2$, show that $\lim_{x\to\infty}u(x)=0$

For $u\in H^s(\mathbb{R}^m)$ with $s>m/2$, show that $\lim_{x\to\infty}u(x)=0$ By the Sobolev embedding theorem $H^s(\mathbb{R}^m)\hookrightarrow C_b(\mathbb{R}^m)$ and this should pretty ...
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1answer
112 views

Extension of uniform limit in Arzela Ascoli

Suppose, I have a sequence $f_n([0, \infty))$ of functions such that they are equicontinuous and uniformly bounded. So, I can get a uniformly convergent subsequence $f_{n_k}$ over $[0,T]$. I want the ...
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1answer
164 views

Generalized Riemann Integral: Bounded Nonexample?

Reference For a convergence theorem on integral see: Riemann Integral: Uniform Convergence For an improper version of integral see: Riemann Integral: Improper Version For a comparison of integrals ...
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24 views

Show that $T(K)\subset K$, that $T$ is an isometry from $K$ to $K$, and that $T$ does not have a fixed point in $K$

Here is my problem: Consider $K=\{f\in L_1[0,1]: 0\leq f\leq 2, \int_0^1fdt=1\}$ and $T:K\to L_1[0,1]$ defined by: $$Tf(t)=\left\{ \begin{array}{l l} min\{2f(2t),2\} & \quad \text{if ...
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1answer
132 views

Are the invertible elements of a Banach algebra closed in the set of left-invertible elements?

Let $A$ be a unital Banach algebra. Denote by $\mathrm{Inv}(A)$ the invertible elements in $A$, and $\mathrm{Inv}_\ell(A)$ the left-invertible elements. That is, $a \in \mathrm{Inv}_\ell(A)$ if and ...
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1answer
46 views

$\|x\|_b=\sum^\infty_{i=1}{1\over 2^i}|x_i|$, is it equivalent to the supremum norm?

Here is my question: Let the $\|\cdot\|_b$ on $c_0$ be a norm defined by: $$\|x\|_b=\sum^\infty_{i=1}{1\over 2^i}|x_i|$$ Is $\|\cdot\|_b$ equivalent to the supremum norm? So I know that if ...
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77 views

Understanding dual spaces

My professor asked me to give him an example about dual spaces from our real life so I was thinking if we take the set of all our actions as the space X can the space of all our thoughts be the dual ...
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1answer
45 views

Prove $\|\cdot\|_s$ is a norm, and find $m,M>0$ such that $m\|x\|_\infty\leq \|x\|_s\leq M\|x\|_\infty$

Here is my question - Let $\|\cdot\|_s:\mathbb{R}^2\to\mathbb{R}^2$ be defined by: $$\|(x_1,x_2)\|_s=\left\{ \begin{array}{l l} \|(x_1,x_2)\|_2 & \quad \text{$x_1x_2\geq 0$}\\ ...
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1answer
226 views

Locally Compact Stone-Weierstrass Theorem

STATEMENT: Let $X$ be a locally compact Hausdorff space, and let $A = C_∞(X)$ be the algebra of continuous real-valued functions on $X$ that vanish at infinity, as above, equipped with the supremum ...
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38 views

Deconvolution of two delta functions (solving $y(t) = A x(t-a) + B x(t-b)$)

I would like to calculate $x(t)$, when only $y(t)$ with $y(t) = A x(t-a) + B x(t-b)$ is known. Since this is a linear shift invariant operation (convolution), the inverse relation must be of the ...
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1answer
40 views

Is the set of unitary matrices compact in finite dimensions?

I am trying to show that the set of unitary matrices is compact in finite dimensions. I began by expressing a non-invertible matrix as a sequence of invertible matrices and then breaking these ...
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33 views

When is the Sturm-Liouville operator $ Lf=x^2f''+xf'$ positive

On the interval $[a,b]$ what conditions make the operator $L= (x^2)D^2 + xD$ positive? here $D$ is the differentiation operator.
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1answer
42 views

Constant for polynomials of limited degree

I'm trying to prove the following: There exists a constant $C \in \mathbb{R}$ such that for all polynomials $f(t) \in \mathbb{R}[t]$ of degree not greater than $ 2014 $ we have $$ |f(10)| \leq ...
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1answer
48 views

Inequality regarding convex combination of random variables

In the appendix of notes on stochastic integration that i am reading, Mazur's Lemma is presented as following http://i.stack.imgur.com/GUyXN.png I have trouble understanding/proving the following ...
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1answer
49 views

$A$ is a symmetric operator ? Please criticize my proof.

Let $A:L^2([0,1])\to L^2([0,1])$ given by $$ Af(t)=\int_0^1K(s,t)f(s)ds, $$ where $K$ is a mensurable square integrable operator, i.e $\int_0^1\int_0^1|K(s,t)|^2\,dsdt<\infty$. $A$ is acompact ...
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56 views

Topological vector space question

$C[0,1]=$ space of all continuous complex valued function over $[0,1]$. Define metric, $d(f,g)={\int_{0}^{1} \frac {|f(x)-g(x)|}{1+|f(x)-g(x)|}}$, for all $f,g\in C[0,1] .$ Let $(C[0,1],\sigma)$ ...
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1answer
62 views

Prove $(S+T)^\times = S^\times +T^\times$.

$T^\times$ and $S^\times$ are the adjoint operators of $T,S\in B(X,Y)$, $X$ and $Y$ normed spaces. $T^\times$ and $S^\times$ are defined on the dual spaces which contain the ranges of $T$ and $S$, ...
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0answers
42 views

Duhamel's formulation of a pde

Let's consider the following initial value problem $$u_t=Lu+F,\,\,\,\,u(0)=u_0$$ with $L$ a spatial operator. Which are the minimal assumptions on $F$,$u_0$,$u$ to have the equivalence of the problem ...
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1answer
48 views

Identifying an elementary mistake

The following argument is wrong, because the conclusion is impossible. Could someone please help me identify the problem(s)? Consider the space $l^2(\mathbb{R})$ of square summable sequences. This is ...
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1answer
42 views

Isomorphy of $C_0(U)$ and an ideal

Let $X$ be a topological space, $Y\subseteq X$ closed and $U:=X\backslash Y$. Then $I_Y:=\{f \in C_o(X)\mid f_{| Y}=0\} \subseteq C_0(X)$ is a closed Ideal. I want to show that $I_Y \cong C_0(U)$. I'm ...
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1answer
26 views

A couple questions on the generalized eigenspaces of a nonself-adjoint compact operator on a Hilbert space

Suppose $A:\mathcal{H} \rightarrow \mathcal{H}$ is a compact (not necessarily self-adjoint) linear operator on a Hilbert space. Suppose $\lambda \in \sigma(A)$ is non-zero. Using the compactness of ...
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563 views

Proof Riesz Representation Theorem (bounded linear functional in Lp)

I have a little problem with this proof (I'm using Royden), can you help me? Let $F$ be a bounded linear functional on $L^p$, $1 \leqslant p \leqslant \infty$. Then there is a function $ge \in L^q$ ...
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1answer
47 views

Is it a closed set of $H^1(\Omega)$?

Let $\Omega\subset\mathbb{R}^3$ an open bounded domain (without holes) with boundary $\partial\Omega$ and let $\Omega_1\subset\Omega$. We consider the domain $\Omega\setminus\Omega_1$, that is, a ...
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1answer
52 views

$C_b(X)$ is non-separable for $X$ non-compact

If $X$ is a non-compact space then prove that $C_b(X)$ is not separable, where $C_b(X)$ is space of all bounded continuous functions on $X$. I was trying like this, but got stuck at middle: Take a ...
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1answer
33 views

extension of a finite dimensional subspace

Let $X$ banach space infinite dimensional. If $v\in X$ such that $v\neq 0$. I wonder if it is possible to find a closed completion for $S=$span $\{ v\}$ in $X$ ie if is possible to find $H$ ...
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38 views

Infimum of radially unbound functional

I am having difficulty following a proof about balls (subsets) of radially unbounded functionals. Let $U$ be a Banach Space. Let the space of admissible controls $U_{ad}\subset U \ne \emptyset$ be ...
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1answer
23 views

Properties about Projection

Let $G$ be an operator on Hilbert space $H$ such that kerG is different from {0}. Let also $P$ be the orthogonal projection onto kerG. My question is : are there some conditions to impose on $G$ ...
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159 views

Weak Convergence and Weak Topology

In discussing weak topology of a normed space $X$, a lemma is given as follows. If $(x_n)$ is a sequence in $X$ converging weakly to $x$, then $x_n$ is bounded. I understand the proof of this ...
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82 views

Continuous Strong-Strong Implies Continuous Weak-Weak

Let $X$ and $Y$ be two Banach spaces and let $T$ be a linear map between $X$ and $Y$. Show that $T$ is continuous strong-strong if and only if $T$ is continuous weak-weak. I can see that $T$ ...
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40 views

Example of a non-linear, non-continuous mapping from $\mathbb R$ onto $\mathbb R$ whose graph is closed [duplicate]

Give an example of a non-linear mapping from $\mathbb R$ onto $\mathbb R$ whose graph is closed although it is not continuous.
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71 views

Two Body Schrodinger Equations

I have a question involving the eigenvalues of a two-body Schrodinger equation. Let $$H=-\frac{1}{2m}\Delta_{x_1}-\frac{1}{2m}\Delta_{x_2}+\frac{e^2}{|{{x_1}-{x_2}}|}$$ over the Hilbert space ...
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2answers
141 views

Self-adjoint operator restricted on a closed subspace

Let $A$ be a self-adjoint operator on a Hilbert space $\mathcal{H}$ (possibly unbounded, densely defined with domain $\mathcal{D}(H)$) and let $S$ be a closed subspace of $\mathcal{H}$, ...
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41 views

Noncommutative version of Littlewood's First Principle

There are definitely noncommutative analogues for Lusin's theorem and Egoroff's theorem (found in Blackadar for example). I'm curious if there is a version of the first principle: Every Lebesgue ...
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1answer
111 views

Show that if $C(K)$ is separable, then $K$ is metrisable, for $K$ compact and Hausdorff

My question is simply as the title states: Let $(K,\tau)$ be a compact Hausdorff (topological) space. Show that if $C(K)$ is separable, then $K$ is metrisable. Firstly, I appreciate that this is ...
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44 views

Dual of a locally convex space

Let $X$ be a normed space. Suppose $E$ is a subset of $ X^*$ (The space of continuous linear functionals). For every $\phi\in E$, define seminorm $p_\phi: X\to [0,\infty)$ such that $p_\infty (x)= ...
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1answer
49 views

Energy dissipation

I've been asked to prove the following, but I don't find the way.. Let $\Omega\subset\left\{0<x_n<a\right\}$ be a subset of $\mathbb{R}^n$ such that it is bounded in the $n^{th}$ coordinate. ...
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66 views

Banach spaces involving time

Let's suppoe $u\in L^2(0,T;H_0^1(\Omega))$ with $u'\in L^2(0,T;H^{-1}(\Omega))$. We know that $$u\in C([0,T];L^2(\Omega))$$. In this result can the set $\Omega$ be the whole $\mathbb{R}^n$ or we need ...
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1answer
105 views

Partial Isometries: Introduction

Attention This question has been modified drastically. It is done so the answer below is still correct. It is done so to allow more specialized threads. Problem How do I deal with partial ...
2
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1answer
71 views

Is $L^1(\Omega)$ isomorphic to $l¹$?

I'm trying to understand a statement in the Brezis book, that $L^1(\Omega, \mu)$ is not reflexive. He divides the problem into two parts: one in which $\mu$ is a non atomic measure and another in ...