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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49 views

Open sets are not relatively compact

The following is a question about the answer given here: I have been trying to prove that if $X$ is an infinite dimensional Banach space and $O\subseteq X$ is an open set such that its closure ...
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1answer
52 views

Proving $u$ is compact whenever $u^\ast$ is

Let $X,Y$ be Banach spaces and let $u: X \to Y$ be a linear operator. Let $u^\ast: Y^\ast \to X^\ast$ denote its transpose and assume that $u^\ast$ is compact. I am trying to prove that $u$ is ...
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1answer
24 views

A question about the definition of $(X\rtimes\Gamma)$-C*-algebra

Here is a quotation in the book "C*-algebras and Finite-Dimensional Approximations": Instead of considering the *-algebra of finitely supported functions from $\Gamma$ to $C(X)$ (C(X) denotes all ...
5
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1answer
66 views

Is $C^{\infty}([0,1])$ a Banach space?

I have read that the answer is no, but I am unable to prove it. Give $C^{\infty}([0,1])$ the metric $$d(f,g) = \sum_{j=0}^{\infty} 2^{-j} \frac{||(f-g)^{(j)}||}{1 + ||(f-g)^{(j)}||}$$ associated to ...
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32 views

Question about the zeros of $\zeta_{H}(s,a) \pm \zeta_{H}(s,1-a)$.

Assume $\zeta_{H}(s,a)$ is the Hurwitz Zeta function. Note that for $a=\frac13,\frac14,\frac16$ the zeros of: $$\zeta_{H}(s,a) + \zeta_{H}(s,1-a)$$ are the same as the non-trivial zeros $\rho$ of ...
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2answers
49 views

Extending Positive Functionals: Linearity

How does regularity provide linearity? Given the full Banach space of bounded functions over a suitable set: $$\mathcal{B}:=\{f:\Omega\to \mathbb{C}:\|f\|_\Omega<\infty\}$$ and a linear subspace ...
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12 views

From essential oscillation to a continuous representative

Let $u$ be a measurable function such that for every(former: a.e.) $x\in \Omega$ there holds for sequences $R_n,\delta_n\to 0$ that$$\omega_n:=ess-osc_{B_{R_n}(x)} u\leq \delta_n. \tag{1}$$ Edit: ...
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53 views

Compactness of the Volterra opelator

The Volterra operator is given as \begin{eqnarray} (Vf)(x)=\int_0^xK(x,y)f(y)\,{\rm d}y. \end{eqnarray} By the Arzelà–Ascoli theorem, $V\colon C^0[0,1]\rightarrow C^0[0,1]$ is compact operator. But, ...
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19 views

Suppose $a$ and $b$ are positive numbers. Prove that for every $\epsilon>0$, $\exists$ $C(\epsilon)$ such that $ab\leq\epsilon a^p+C(\epsilon)b^q$.

Suppose $a$ and $b$ are positive numbers. Prove that for every $\epsilon>0$, there is a constant $C(\epsilon)$ such that $ab\leq\epsilon a^p+C(\epsilon)b^q$ where $\frac{1}{p}+\frac{1}{q}=1$. I'm ...
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46 views

Weak*convergence Problem

I have the following exercise: $\pi_n : \ell^{\infty}\rightarrow \ell^{\infty},x\mapsto \pi_n(x)$, where $(\pi_n(x))_k=x_k$ for $0\leq k\leq n$ and $(\pi_n(x))_k=0$ for $k>n$. We claim that the ...
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1answer
40 views

Do $\mathbb{R}^n$ and $\mathbb{C}^n$ valued ordinarily measureable functions form a Banach space under p-norm?

By measureable function I mean an "ordinarily" measureable function, that is measureable in a sense of this definition: a function between measurable spaces is said to be measurable if the preimage of ...
2
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1answer
35 views

Is a bounded Borel function of a normal operator normal

I am playing around with Borel functional calculus to try to understand it, and made the following argument: Let $T\in B(H)$ (bounded operator on Hilbert space) be normal. Let $f\in C(\sigma(T)) $ ...
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13 views

Zigmund-Besov Spaces and Inverse Function Theorem, is the Inverse Zigmund?

Preliminary Definition Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are allowed ...
4
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1answer
95 views

Spectrum of an integral operator.

For any $f\in C([0,1],\mathbb{R})$ set $$ Tf(x) = \int_0^1 [\min\{x,y\}\cdot f(y)]dy. $$ I have just proved that $T$ is a compact operator from $C([0,1],\mathbb{R})$ into itself. I would like to know ...
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1answer
28 views

How to find a sequence in discrete group

Let $\Gamma$ be a discrete group. Can we find an increasing sequence $F_{n}\subset \Gamma$ of finite subsets, such that $\cup F_{n}=\Gamma$?
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1answer
80 views

Extreme points of the unit ball of $l^1(\mathbb{Z^+})$ and $L^1[0,1]$

Determine the extreme points of the unit ball of $l^1(\mathbb{Z^+})$ and $L^1[0,1]$. My attempt: I know the definition but I don't know how to find these extreme points.Please help me to solve this ...
2
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0answers
25 views

Comparison and maximum principle for parabolic pde

I was told the comparison principle can also be understood as the fact that the difference between a subsolution and a supersolution satisfies the maximum principle. I also know comparison principle ...
2
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1answer
33 views

If $X$ is a LCHS and $K, O \subseteq X$ with $K$ cpt & $O$ open, then $\exists U$ open s.t. $K \subseteq U \subseteq \overline{U} \subseteq O$?

I'm having trouble fully understanding the proof of this statement. Suppose $X$ is a locally compact Hausdorff topological space. Then if $K$ is a compact subset of $X$ and $O$ is any open subset ...
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0answers
22 views

Weak separability.

How I can show the following statement? Let $E$ be a normed space and $A\subseteq E$. Then $A$ is separable if and only if $A$ is weak-separable. If $A$ is separable, is clear that $A$ is ...
3
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1answer
53 views

Give necessary and sufficient conditions for a multiplication on $L^p$ to be compact

Let $(X, \Omega, \mu)$ be a $\sigma-$ finite measure space and for $\phi \in L^\infty(\mu)$ let $M_\phi:L^p(\mu) \to L^p(\mu)$ defined by $M_\phi f = \phi f $ be the multiplication operator. Give ...
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58 views

Volterra operator and completely continuous operators

Consider the Volterra operator $V$ defined here. Let me give some definitions first: [Dunford-Pettis] We say that a bounded linear operator $D:L_1[0,1]\to L_1[0,1]$ is Dunford-Pettis if it sends ...
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0answers
29 views

Is Sobolev regularity propagated under evolution?

Given a well-posed initial problem in a domain $\Omega$ of the form: \begin{equation} \square\phi=f \end{equation} where $\square$ is the wave operator, $f\in L^{2}(\Omega)$, with initial ...
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0answers
26 views

Prove the property of the theorem

Before you submit theorem for which I am concerned will first give some necessary data $1)$ Proposition 1: Let be $X$ a normed space, $Y$ unitary space and $A:X\rightarrow Y$ linear operator. The ...
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1answer
16 views

subadditivity and continuity at zero implies continuity at all points

If $p$ is a subadditive functional on a normed space $X$ and is continious at $0$ and $p(0)=0$. To show $p$ is continious for all $x \in X$. This is a problem from Kreyszig's Introductory Functional ...
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0answers
19 views

integration formula

help me please all these functions are regular. How we can found this formulation $$ \displaystyle\int_{\Omega} (f(u)-f(k)) \nabla p(g(u)-g(k)) \xi dx = - \displaystyle\int_{\Omega} H(u,k) \nabla \xi ...
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votes
2answers
46 views

Bochner: Absolute Integrability

For a Bochner measurable function it holds: $$f\text{ Bochner integrable}\iff\|f\|\text{ Bochner integrable}$$ for any positive measure $\lambda\geq 0$. The one direction is relatively simple when ...
3
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1answer
46 views

If $T$ is self-adjoint, is the set of power series in $T$ closed?

If $T$ is a bounded self-adjoint operator on a Hilbert space, is the set of convergent power series in $T$ closed in the norm topology? I ask because I'm reading some spectral theorems and I was ...
2
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1answer
22 views

An abstract a priori estimate in finite element method

Let $V$ and $K$ be Banach spaces (with norms $\|\cdot\|_V$ and $\|\cdot\|_K$ resp.) and suppose that there is a compact linear embedding $K\hookrightarrow V$. Furthermore, let $P_n$ be a family of ...
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1answer
25 views

Extending Banach Space of Functions

The idea is that one could in principle consider the space of functions: $$\{f:\Omega\to V\}$$ with pointwise operations and uniform convergence: $$f_\lambda\to f:\iff\|f_\lambda-f\|_\infty\to 0$$ ...
1
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1answer
71 views

Misunderstanding about Laplace operator

Let $\Omega$ be a bounded subset of $\mathbb{R}^n$. We know that the Laplace operator \begin{align} \Delta \colon H_0^1(\Omega) \to L^2(\Omega) \end{align} admits an inverse operator \begin{align} A ...
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0answers
35 views

Is every subspace of a normed linear space which is not closed a hyperspace.

Let $B \subset X$ where $X$ is a normed linear space over $\mathbb{R}$ and $B$ is a proper subspace. If $B$ is not closed, is $B$ necessarily a hyperspace(maximal proper subspace) in $X$. I attempted ...
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2answers
47 views

It's the discrete topology.

I have to proof that if I have $(X,\tau)$ and $(Y,\delta)$ two topological spaces, if every function $f:X\longrightarrow Y$ is continuos then $\tau$ is the discrete topology. I don't know what is the ...
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0answers
18 views

numerical+variational mixed optimization $\min_{l,u,f(.)} (u-l)+\int_{-1}^1 (f'(x)-x)^2dx$

Given a function $g(x)$ and its domain, we want to get another function $f(x)$ whose derivative approximate $g(x)$ well, but so that $f(x)$ itself has small variation.for example, for ...
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0answers
26 views

Radon-Nikodym: Complex Measures

Let $\Omega$ be a measureble space and $\mu$ a complex measure. (Note that this implies that the measure is finite.) Consider an absolutely continuous complex measure $\nu\ll\mu$. Then: $$\nu=\int ...
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1answer
23 views

About a Property of maximal solutions of separable ODE's $y'=g(x)h(y)$ for locally Lipschitz $h : U\to\mathbb R$, $U$ open

Theorem: Let $\varphi : (a,b) \to \mathbb R$ be a maximal solution of the IVP $$ y'(x) = g(x) \cdot h(y(x)), \quad y(x_0) = y_0 \quad (1) $$ with continuous functions $g : I \to \mathbb R$ and $h : U ...
2
votes
0answers
53 views

Dual subfactor and commutant

Let $(N \subset M)$ be a subfactor and $N \subset M \subset M_1$ the basic construction. Question: Is $(M \subset M_1) \simeq (M' \subset N')$? Else in which generic case it's true? What's the ...
2
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0answers
37 views

Riesz-Markov-Kakutani Theorem: Various Versions

The Riesz-Markov-Kakutani theorem usually comes in various versions. So I'm a little bit confused and wondering which of these are right. Let $\Omega$ be a locally compact space. Then: Complex ...
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1answer
34 views

Question about proving a function on linear space to be norm

Let $X$ denote the linear space of all polynomials in one variable with coefficients in $\mathbb{K}$. (Where $\mathbb{K}$ denotes $\mathbb{C}$ or $\mathbb{R}$). For $p \in X$ with $p(t) = a_0 + a_1t ...
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0answers
15 views

Solution set of linear operator equations

Suppose $\mathcal{X}$ and $\mathcal{Y}$ are two Hilbert spaces. Let $A:\mathcal{X} \mapsto \mathcal{Y}$ be a bounded linear operator. Consider a linear operator equation $Ax=b$. My question is what ...
2
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1answer
41 views

Three-space property

I have found two definitions of a three-space property. One definition is: $(P)$ is a three-space property if whenever $E$ Banach space, $F\subseteq E$ is a closed linear subspace and two of the ...
2
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1answer
34 views

Markov processes on function spaces

Is there any reference on Continuous time Markov process whose state space is infinite dimensional function spaces, such as the space of continuous functions $C(R^d)$? It seems Dirichlet Form is a ...
2
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1answer
20 views

$\sup$ norm of a function

The following is an example of Murphy's C*-algebras and operator theory: I do not know how he concludes $$\int_0^1 |k(s,t) - k(s',t)||f(t)| dt \leq \sup|k(s,t) - k(s',t)|||f||_\infty$$ Please help ...
2
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0answers
46 views

Prove that a norm makes a space Banach

I have to prove that if $A$ is a C*-Algebra then the algebra $A_1$ obtained adjoining the identity is a C*Algebra too (with the usual algebraic operation defined). I have any problem in all the ...
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0answers
35 views

when does classical theory of parabolic PDE fail

I am reading a paper on degenerate parabolic PDE and I am confused about the following statement. " Diffusion coefficient is $mu^{m-1}$, and it vanishes when $u=0$. Hence at all those points where ...
0
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1answer
38 views

Periodic Laplace operator non closed in $ C^2(0,L)$

How can I show that the Laplacian operator is not closed in the domain $D=\{f \in C^2(0,L) \mid \mbox{ f is vanishing in a neighborhood of 0 and L } \}$ for a fixed $L$? And how can I show that it is ...
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1answer
43 views

proving Orthonormal basis

I have given a set of functions in $L^2\left(\left[-\frac{a}{2}, -\frac{a}{2} \right]\right)$ consisting of the following functions: $$u_{n}(x)=\sqrt{\frac{2}{a}}f_n(x),$$ where $f_n(x)= ...
3
votes
1answer
52 views

Integral operator on $L^p$ is compact

Let $(X,\Omega,\mu)$ be an arbitrary measure space, $1<p<\infty$ , and $\frac{1}{p}+ \frac{1}{q} = 1$. If $k:X. X\to \Bbb C$ is an $\Omega.\Omega-$ measurable function such that $$M = [\int ...
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0answers
34 views

Regarding metrizability of weak/weak* topology and separability of Banach spaces.

Let $X$ be a Banach space, $X^*$ its dual, $\mathcal{B}$ the unit ball of $X$ and $\mathcal{B}^*$ the unit ball of $X^*$. The following result is well-known Theorem. If $X$ is separable, then ...
4
votes
1answer
48 views

Is there $u,v\in L(E): uv-vu=id_E$

Let $E$ be a normed vector space over $\mathbb{R}$. Is there continuous linear transformations $u$ and $v$ such that: $$uv-vu=id_E$$ (.ie $\forall x\in E:u(v(x))-v(u(x))=x$) I suspect that the ...
2
votes
3answers
111 views

Volterra operator is completely continuous

Let $\mu$ be the Lebesgue measure on $[0,1]$ on the borelians, and consider the Volterra operator $V:L^1[0,1]\to C[0,1]$ given by $$ Vf(t)=\int_0^t f d \mu $$ So, I want to show the following ...