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 question on Holder spaces

A function $f$ is said to belong to the Holder space if Holder condition is satisfied, i.e. $\exists \beta,L\geq0$ such that $$|f(x)-f(x')|\leq L|x-x'|^\beta$$ for all $x,x'$ in the domain of $f$. ...
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33 views

Confusion with proving that some subspace of a Banach-Space is closed

So here is my problem, I am trying to show that, Let $X,Y$ be Banach-Spaces and $T:X\rightarrow Y$ a linear and bounded map. Then $T(X)$ is closed if $Y/T(X)$ is of finite dimension. While ...
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52 views

Scalar products and partitions of Hypercubes

My questions relate to scalar products defined in $\mathbb{R}^{n}$ and partitions of hypercubes. Take $s \in \mathbb{R}$, $\xi, \eta \in \mathbb{R}^{n}$. My first question is why is it possible to ...
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25 views

Question about subadditive functionals

Def: Let $X$ be a vector space. $f$ is called a subadditive funcional in $X$ is $f$ is a functional(real - valued) and it satisfies : $$f(x + y ) \leq f(x) + f(y) \; \; \; \forall x,y$$ Question: ...
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151 views

Strongly convex and Frechet differentiable function in reflexive Banach space

We first recall two definitions about strong convexity and Frechet differentiability in normed space. Let $(X, \|.\|)$ be a normed space and $f:X\rightarrow\mathbb{R}$ be a function. (a) $f$ is said ...
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2answers
389 views

dual of $H^1_0$: $H^{-1}$ or $H_0^1$?

I have a problem related to dual of Sobolev space $H^1_0$. By definition, the dual of $H^1_0$ is $H^{-1}$, which contains $L^2$ as a subspace. However, from Riesz representation theorem, dual of a ...
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276 views

Integration in Banach spaces - interesting, nice and non-trivial examples needed

I am interested in $\textbf{Integration in Banach spaces}$. Here is a little motivation for my question: Let $\left(X,\|\cdot\| \right)$ be a Banach space, $a,b \in \mathbb{R}$ with $a<b$ and $f ...
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1answer
79 views

I have to decide if $ \ell^1\subset c_0$ is closed or not.

I was asked to decide if $ \ell^1\subset c_0$ is closed or not, where $$\ell^1=\{(x_n)_{n\in\mathbb N}\subset\mathbb R:\sum_{n=0}^{\infty}|x_n|<\infty\}$$ $$c_0=\{ (x_n)_{n\in\mathbb ...
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1answer
141 views

Weak*-convergence of probability measures

Let $(\Omega,\mathcal F)$ be a measurable space and $X:\Omega\rightarrow\mathbb R^d$ a $\mathcal F$-measurable, bounded map. Let $(\mathbb Q_n)_{n\in\mathbb N}$ be a sequence of probability measures ...
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66 views

Linear functional and a bounded norm

I'm trying to work through some example questions for my class but there is no mention of "Piecewise affine functions" anywhere and I'm completely stumped on how to do this. Any help would be ...
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1answer
65 views

Basic sequence- what is so special about it?

Let $(x_n)$ be a Schauder basis of a vector space $X$. This means that the $span(x_n)$ is dense in $X$, right? Then wikipedia introduces the notion of a $\textbf{basic sequence} $ when $(x_n)$ is a ...
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61 views

Spectrum and characters: could anyone please check my proof

I tried to prove the following: Let $A$ be a commutative non-unital complex Banach algebra and $\chi : A \to \mathbb C$ a character. Then $$ \sigma (a) = \{\chi (a) : \chi \in \Omega (A) \} ...
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61 views

Don't understand a PDE argument involving $L^p$ norms and inequalities

I'm reading this paper. I do not understand how the author proves Corollary 5.12 for the case $p=2$. He addresses everything except $p=2$ but claims it holds for all $p$. Can someone help me to see ...
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38 views

The uniqueness of a solution of a system of differential equations

Suppose I have a system of delay differential equations $\frac{d G_1(\phi_1(s))}{ds}= F_1(\phi_1(s),\phi_2(1-s),1-s)$ $\frac{d G_2(\phi_2(1-s))}{ds}= F_2(\phi_1(s),\phi_2(1-s),s)$ where $G_1, G_2, ...
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2answers
325 views

Borel Sigma Algebra

The question is asking to prove that the family: $\{(−a, a) : a \in \mathbb{R}\}$ does not generate the Borel $\sigma$-algebra. It is known that the family $\{(a,b) : a < b\}$ generates the Borel ...
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1answer
88 views

subspace of Hilbert space is closed if and only if it is weakly closed

Any hints for this question, thank you! Prove that a subspace of Hilbert space is closed if and only if it is weakly closed.
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3answers
65 views

Gel'fand representation of a non-unital Banach space: what's wrong with this argument

My argument below is hacked together from pages 5-6 of Davidson's "$C^*$ algebras by example". Theorem: The multiplicative linear functionals on a unital abelian Banach algebra are continuous of ...
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2answers
46 views

Bounded linear operator and self adjoint operator

these questions are in my workbook but there is no worked solutions whatsoever. I dont know where to begin with this at all. Any help would be much appreciated. Thankyou
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1answer
74 views

convergence in L^{1} strong

I search an proof of this lemma: First,we have this definition: we tell that an sequence $f^{\epsilon}$ is equi integrable if $$\forall \eta, \exists \delta > 0, |E|\leq \delta \implies ...
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1answer
109 views

Why is the topology of convergence in measure equivalent to this metric here?

I am currently struggeling with the topic of convergence in measure topologies. Now I read that on the space of measurable function $L^0$ on $[0,1]$ with the Borel sigma algebra and the lebesgue ...
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1answer
49 views

Which of the following sets are open (or closed)?

a.) $A:= \{(x_n)_{n\in \mathbb{N}} : x_n \in [0,1] \hspace{2mm}\text{for all}\hspace{2mm} n\in\mathbb{N}\}$ in $(l^\infty, \|\cdot\|_{\infty})$ and b.) $B:= \{f\in C([0,1]) : |f(t)-t|<1 ...
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1answer
31 views

An identity on direct sum of Hilbert spaces

Let $M_i$ are the set of smooth complex valued functions ($i=0,1,2,...$) $L^2(M_i)$ are Hilbert spaces on $M_i$ then can we say $$L^2(\bigoplus_{i=0}^\infty M_i)\cong \bigoplus_{i=0}^\infty ...
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52 views

Is this condition sufficient to determine the linear space is of finite dimension?

From the Banach theory we knew that: 1) A linear space(a vector space endowed with its vector topology) $X$ of finite dimesion $dimX=n$ has the following property: If ${\left\| \bullet \right\|_1}$ ...
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1answer
30 views

Density transformation, distribution function

Suppose $X$ is a real-valued random variable with density $p_X$, so $$P(X\leq x) = \int_{-\infty}^x \, p_X(y) \, dy.$$ What conditions on a function $f$ are needed (typically?) to find the density ...
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227 views

Gilbarg Trudinger: Hölder continuity in chapter 8

I'm trying to track the behaviour of the coefficients in Theorems 8.22 and Theorem 8.24. Particularly, I'm considering the behaviour w.r.t. to the distance from $\Omega'$ to $\partial \Omega$ I'll ...
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49 views

What can be said about this family of functions?

$$\sqrt{a^2+x^2+\left(y-\frac{b}{2}\right)^2}-\sqrt{a^2+x^2+\left(y+\frac{b}{2}\right)^2} = \frac{(2n+1)\pi}{k} \space\space\space\space n \in \mathbb{Z}, \space\space a,b,k > 0$$ This has popped ...
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1answer
36 views

Lebesgue integral for unbounded domain.

Suppose that $\Omega$ is unbounded domain. So i want to know the Lebesgue integral on $\Omega$. Detail: $$\int\limits_\Omega d\mu=?$$ I think the result is $|\Omega|$. Is it true?
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37 views

Question about special $C^*$-algebra

i have a question about a $C^*$ algebra $A$ namely $M_2(\mathbb{C})$. I want to prove that every state $\alpha$ of $M_2(\mathbb{C})$ (thus a positive linear functional with norm $1$) is of the form ...
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58 views

Can a separable Banach space have a total subset?

I will like to know whether a separable Banach space can have a total subset? A separable Banach space has a countable dense subset V, but can we do something to make V total? Thank you for your ...
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25 views

Proving differentiability of simple polynomial map

I'm trying to show the differentiability of the following simple polynomial map $\phi$ $$\phi : P \mapsto P^3, P \in \mathbb{R}_q[X]$$. I go by the definition of differentiability, i.e. I search for a ...
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1answer
58 views

Equivalent conditions for composition to be compact operator

I did some exercises in Conway's functional analysis book and found the following problem: Let $\tau:[0,1]\to [0,1]$ be continuous and define $A:C[0,1]\to C[0,1]$ by $Af:= f\circ \tau$. Give ...
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11 views

invariant measures for the independently coupled Feller process

If I take countably many Markov processes and couple them independently, how does the collection of invariant measures for the coupled process relate to those of the marginals? I conjecture that it ...
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1answer
30 views

Norm of a bounded and linear functional

Let $X$ be a normed space and $x\in X ~,~f\in X^*$. I do not know why is There a sequence $\{x_i\}\subset X_{||.||\leq 1}$ such that $||f||=\lim|f(x_i)|$. Please help me
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48 views

what are closed sets in $L^{1}(\mathbb R)$?

Consider, $L^{1}(\mathbb R)$= The space of Lebesgue integrable functions on $\mathbb R$; for $f\in L^{1}(\mathbb R),$ we define its norm, by $\|f\|_{L^{1}}=\int_{\mathbb R}|f(x)| dx$; It is well-known ...
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1answer
83 views

Bounded continuous functions from a topological space to $\mathbb R$ is complete

Let $X$ be a topological space and let $BC( X \to \mathbb R)$ be the space of bounded continuous functions from $X$ to $ \mathbb R$ equipped with the supnorm $||*||_{\infty}$. How to prove $BC( X \to ...
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1answer
55 views

Linear functional in Banach space

Let $X$ be a Banach space, $(f_n)\in X^{*}$ a sequence with $f_n\neq 0 $ $ \forall n\in \Bbb N$. Show that there is a $x\in X$ such that $f_n(x)\neq 0 $ $\forall n$. Need some help. Thank you!
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Biorthogonal functions in $L^p$

I asked one question that is already answered: 1.) I have a question about Lemma 9.5 on page 93/94 reference. It's about the part of the proof where the sequence of $(g_n^*)$ are introduced. I don't ...
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1answer
53 views

$(\ell_p,\|.\|_{\infty})$ Banach or separable

Is $(\ell_p,\|.\|_{\infty})$ for $1\leq p<\infty$ a Banach or separable space? is there any fast way of proving it without checking separability or completeness with the usual way?
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189 views

Prove there is no strictly increasing function $f$ from irrationals to reals.

Prove there does not exist a strictly increasing function $f\colon \mathbb{R} - \mathbb{Q} \rightarrow \mathbb{R}$ such that $f(\mathbb{R} - \mathbb{Q}) = \mathbb{R}$. I imagine the best way to go ...
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0answers
78 views

Expectation of $p$-norm under a Gaussian on the Hilbert space $L^2(S^1)$

Let $\mu$ be a centered Gaussian measure with (nondegenerate) covariance $Q$ on the Hilbert space $L^2(S^1;\mathbb R)$ where $S^1$ is the circle. We can take for example the covariance ...
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51 views

Analytic solutions of an equation

I am trying to find analytic solutions of this equation for $x$ with parameter $a$ ($x>0, a>0$): ...
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1answer
43 views

Sobolev inequality in negative index

For $s>n/2$, is it true that $$ \int |fg| dx\leq ||f||_{H^s}||g||_{H^{-s}}?$$ This inequality is used on pg 398 of the Majda Bertozzi book on Vorticity and Incompressible flow but I can't make ...
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308 views

Properties of the square norm in Banach spaces

Let $X$ be a Banach space with its dual $X^*$. Consider the mapping $f: X\rightarrow \mathbb{R}$ given by $$ f(x)=\frac{1}{2}\|x\|^2. $$ We have know that when $X$ is a real Hilbert space ($X=X^*$) ...
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1answer
23 views

equivalency of weak convergence and strong convergence for this family of sequences

Let $H$ be a Hilbert space and let $f_n \in H$ be a sequence of orthogonal elements i.e $<f_n,f_m>=0 $ if $n\ne m$. Define the element $F_N= f_1 + f_2 +...+ f_N$ for each $ N\in \mathbb N$. ...
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28 views

Schauder's theorem: consequences and applications

I am about to give an informal talk about Schauder's theorem ($T:X\to Y$ linear operator between Banach spaces is compact if and only if its adjoint is). Does anyone know any derived ...
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2answers
91 views

Dense Countable basis on Hilbert space

Let say that I have a $H$ hilbert space and linear independent countable set $\beta =\{ \beta_1 , \beta_2, \beta_3... \}$ such that $span(\beta)$ is dense set in H. does $span(\beta-\beta_1) =span( ...
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43 views

Question on Morse inequalities

I want to understand why: if i have then $(4.1)$ is formal : it means that please help me Thank you EDIT1: $(4.1)$ tel us that $\displaystyle\sum_{q=0}^{\infty} (M_q-\beta_q)t^q=(1+t)Q(t)$ ...
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197 views

Is this Neumann series solution unique?

I have a Fredholm integral equation of the second kind given as $$f(x)=g(x)+\lambda\int_{-\infty}^\infty K(x,y)f(y)dy, $$ where $\lambda\in(0,1)$, the kernel $K(x,y)=\phi(x-y)$ is a Gaussian ...
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60 views

The infinitessimal generator of Brownian Motion

Background: I know, and can almost prove, that the infinitessimal generator of BM is the Laplacian/2. For me (and I have never heard it done differently) we have Feller Processes, of which BM is one ...
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263 views

On the weak convergence in reflexive Banach space

Consider the following proposition: Proposition 1. Let $X$ be a reflexive Banach space and suppose that $\{x_n\}$ is a sequence in $X$ that is bounded and has at most one weakly sequentially cluster ...