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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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 ...
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37 views

Motivation for the notion of locally convex topological vector space

Is the only motivation for the notion of locally convex topological vector space that the local bases have some nice property i.e. convex, balanced, absorbing ?
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42 views

Inequality $\Big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\Big\|_{H^1(\mathbb R)} \le C\|f(x)\|_{L^\infty(\mathbb R)}^2 \|f(x)\|_{H^1(\mathbb R)}$

For complex-valued functions $f_1, f_2, f_3:\mathbb R\to\mathbb C$, I want to know that the following inequality holds: $$ \Big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\Big\|_{H^1(\mathbb R)} \le ...
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67 views

Absolute continuity and convolution

Suppose that $\mu$ is a finite Borel measure on the real line, $f, g\in L^1(\mu)$. Define $\nu=\mu\ast\mu$. Do I understand correctly that the convolution $f\mu\ast g\mu$ is absolutely continuous wrt ...
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50 views

Bounded below adjoint operators

Let $T\colon X\to Y$ be a bounded linear operator. Suppose that $Z$ is a subspace of $Y^*$ such that $T^*$ is [bounded below][1] on $Z$. Denote by $\text{w*-dens}\, Z$ the minimal cardinality of a ...
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34 views

Does a strictly convex and weak metrizable unit sphere of a Banach space imply separability?

I want know If $X$ be a Banach space with strictly convex norm and a metrizable unit sphere, $S_X=\{x\in X: \|x\|=1\}$, for the weak topology. Does a strictly convex and weak metrizable unit ...
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81 views

Existance of inverse of an operator

I was studying abstract inverse source problem of an abstract heat equation in approach of semi group theory. There I am unable to find the reason of existence of inverse of an operator that i have ...
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64 views

Inequality proof using the triangle inequality

I am reading Kreyszig's Intro to Functional Analysis and am a bit stoked with one of the problems (problem 12 in section 1.1, page 9): Problem: Given a metric space $(X, d)$, show, using the ...
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87 views

Dimension of the range of an operator and its adjoint

Proof for $\dim(R(T))=\dim(R(T^{*}))$ for a linear operator in a Hilbert space. $T$ is the operator and $T^{*}$ is its adjoint. I would like to know about the authenticity of the following line ...
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54 views

Clarification of a theorem from Chang's Methods in Nonlinear Analysis

The following theorem is taken from Chang's Methods in Nonlinear Analysis. It has a complete proof; however, I have some trouble understanding it (for example, I don't see what $K(f_{\sigma_i})$ ...
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46 views

A proof for Jensen’s inequality

I’m trying to prove a version of Jensen’s inequality, but I end up with the wrong result. I’d appreciate any help or comments. The theorem states: let $\varphi :{{R}^{k}}\to R$ be convex. Then for ...
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32 views

On characterization of Riesz homomorphisms on $C(X)$ space

How to prove the following: Let $K$ be an arbitrary topological space and $\pi: C(K)\to\mathbb R$ be a map with $\pi (1) = 1$. If $\pi$ is a algebra homomorphism then it is an Riesz homomorphism.
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18 views

Resolvent set of extension

Suppose X is an incomplete normed space and X' is its completion. Let T be a bounded linear operator on T and T' the linear extension of T to X'. How do I prove that the resolvent set p(T) of T ...
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31 views

Uniform continuity of weighted Sobolev functions.

I am trying to show an embedding result for a weighted Sobolev space and have come to the following problem: I have a function $f: (0,a] \rightarrow \mathbb{R} $ such that: $f$ is bounded and ...
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53 views

Prove that operator is completely continuous

Let's consider Banach space $\ell^\infty$ of bounded sequences $x = \{ \xi_n\}_{n=1}^\infty$: $$ ||x|| = \sup_{n \in \mathbb N} |\xi_n|. $$ Suppose matrix $||a_{i j}||_1^\infty$ specifies operator $A$ ...
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61 views

Fourier transform and inverse transorm

I have to prove that for $f\in L^1(\mathbb{R})$ $$ \check{\hat{f}}=\hat{\check{f}}, $$ where $$ \hat{f}(\xi):=\int\limits_{\mathbb{R}}e^{-i\xi x}f(x)\mathrm{d}x $$ and ...
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40 views

amazing boundedness problem from maximal function

Let $n\geq 2$. For any $M>1$, prove that there exists a constant $C_M>1$ such that for any ball $B$ in $\mathbb{R}^n$, if we denote $MB$ as the concentric ball of $B$ with $M$ times radius of ...
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42 views

Elliptic regularity on the torus: reference request

Suppose we work on the two dimensional torus $\mathbb T^2$. Let $L_a^2$ be the space of square integrable functions with zero space average and $H_a^m$ be the corresponding Sobolev space. Suppose we ...
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159 views

Show that an hyperplane is closed iff f is linear and continuous

I need an help with the following exercise. Let $(E,\| \cdot \|)$ a n.v.s. and let $f:E\rightarrow \Bbb R$. Show that $H=\{x\in E: f(x)=\alpha\}$ is closed if and only if $f\in E'.$ Actually, I ...
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44 views

basic sequence in the complexification induces a basic sequence in the underlying real space?

This should be easy to prove if it is true, but, alas, what SHOULD be easy is not always easy for me ;) Conjecture 1. Let $X$ be a real Banach space and let $X_\mathbb{C}$ denote its ...
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24 views

Show P is linear if it is convex and positive homogenious functional

it might be too simple but I couldnt show the second part L linear real space $P:L\rightarrow \Bbb R$ is called positive homogenious functional if for every $x\in L$ and $\alpha\ge 0$ , $P(\alpha ...
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46 views

Is the plane created by ($\int f_0^u f_1^{1-u},\int f_1^u f_0^{1-u})$ continuous?

$f_0$ and $f_1$ are two continuous density functions on $\mathbb{R}$. I wonder if $$(x,y):=\Bigg(\int f_0^u f_1^{1-u},\int f_1^u f_0^{1-u}\Bigg)$$ for all $f_0$ and $f_1$ is complete for some ...
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23 views

Question concerning the spectrum of an element of $\mathscr{l}^1(\mathbb{Z})$

I have the following question: $A=\mathscr{l}^1(\mathbb{Z})$ is a Banach Algebra using convolution as multiplication. I now would like to know, if there is a nice way to express the spectrum ...
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36 views

How to show if $M$ is convex set then $\ker(M)$ is convex?

Let $L$ be real linear space and $M\subset L$ is convex. Show that $\ker(M)$ is convex $\ker(M):=\{x:x\in M$ for all $y\in L$ there is $\epsilon = \epsilon(x,y)>0$, for every ...
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129 views

Can we characterize homogeneous Sobolev spaces by means of Sobolev embedding?

For $\alpha>0^{[1]}$, let $D^\alpha=(-\Delta)^{\frac{\alpha}{2}}$ be the operator defined on all Schwartz class functions $f$ as follows: $$ \widehat{D^\alpha f}(\xi)=\lvert \xi\rvert^\alpha ...
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50 views

Proving that a Sturm-Liouville problem is in the limit-point/-circle case

I would like to understand techniques anybody is able to detail to me on how one may actually prove that a particular Sturm-Liouville (S-L) problem, i.e., of the form \begin{equation} ...
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57 views

Applying the Hahn-Banach separation theorem

I have a question applying the Hahn-Banach theorem. I would apply this version of the Hahn-Banach separation theorem. Theorem. Let $V$ be a topological vector space over $\mathbb{R}$. If $A$, $B$ are ...
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41 views

can we expect a bounded approximate identity in $L^{1}(\mathbb R)$, whose Fourier transform at particular point is zero?

Let $\phi:\mathbb R \to \mathbb C$ be a function; and define $$\phi_{t}(x):=\frac{1}{t}\phi (x/t), (t>0).$$ We let, $\phi\in L^{1}(\mathbb R),$ then $\phi_{t}\in L^{1}(\mathbb R);$ in fact, we ...
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49 views

Basis representation for non-negative, compact support, reasonably smooth spectral function

I was wondering if anyone has ideas on representing a non-negative, compact support (from x=-1 to 1 on the real axis) spectral function as a superposition of basis elements. Ideally, the basis ...
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104 views

Properties of the essential supremum

In the article about testability of densities (Devroye and Lugosi 1999, page 4) (link: http://repositori.upf.edu/bitstream/handle/10230/1024/375.pdf?sequence=1 ). They use some properties of the ...
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30 views

a (probably trivial) question on trace theory

I was going through the trace theorem on Sobolev space which speaks of its existence for function $\in W^{k,p}(\Omega)$. My question is whether the trace operator unique if it exists?. My intuitive ...
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45 views

About a property of weighted Sobolev spaces

If we define the weighted Sobolev norm as $$ \|f\|_{H^{s,k}(\mathbb R^n)} := \|(1+|x|^2)^{k/2} (\sqrt{1-\Delta})^s f \|_{L^2(\mathbb R^n)} $$ where $f(x):\mathbb R^n \to \mathbb R$ and $\Delta$ is ...
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87 views

Functional equation relating to normal numbers

My coauthor and I have run into the following problem in a research project involving normal numbers. We suspect that the following question may be resolved using standard techniques in analysis. We ...
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87 views

Use a fixed point argument to show there exists a unique solution to the following BVP

Show using a fixed point argument that there exists a unique solution $f\in C[0,1]$ to $$ -f''(x)+\sin(f(x))=\sin(x) , x\in (0,1), y(0)=y'(1)=0 $$ This is what I have so far: We can show ...
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46 views

Eigenvalues of rotation invariant operators on 2-sphere

Work on $L^2(S^2)$, where $S^2$ is the 2-sphere. Suppose that I have an operator, $T$, that is rotation invariant. That is, $T$ commutes with $R$ for any rotation operator $R$. Suppose furthermore ...
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43 views

The definition of Hardy space

The definition of the Hardy space consists functions $u(z)$ that are analytic a. outside the closed unit disc or b. inside the open unit disc What is the difference between the two definition ...
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62 views

Application of a general “Weierstrass theorem”

http://books.google.at/books?id=9OSrV73a40gC&pg=PA45&lpg=PA45 gives a general Weierstrass theorem. Are there notable applications of this theorem, say in the calculus of variations? (I could ...
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73 views

Can complex-valued affine function be approximated?

If $K$ is a compact convex set of a locally convex Hausdorff space $V$ over $\mathbb{R}$ and $A(K)$ is the set of all continuous affine real-valued function on $K$, then the set of all restrictions to ...
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33 views

boundedness of $\{\int_E(g f_n)\}$ implies boundedness of $\{f_n\}$

I need some help on this problem: Let $E$ be a measurable set, $1 \le p < \infty$ and $q$ is the conjugate of $p$. Suppose that $\{f_n\}$ is a sequence in $L^p(E)$ such that for each $g \in ...
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22 views

Equivalent Gaussian measures

Let $\mu$ be a gaussian measure with eigenpair $\{e_k,2^{-k}\}$ and $\nu$ with eigenpair $\{ Te_k,2^{-k}\}$. Here, $T$ is the unitary operator given by $Tx = x - 2\left\langle x,v \right\rangle v$. ...
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50 views

Linear w.r.t. any measure

Let $X$ be a Banach space endowed with a Borel $\sigma$-algebra. How do we call a real-valued Borel function $f$ that satisfies for any Borel probability measure $\mu$ the following formula $$ ...
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61 views

Real Hardy space question

Please see this related question for the definition of the grand maximal function and the class of normalised test functions $\mathcal{T}$. I will refer to them in this question. Let $f\in ...
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61 views

Representation of a linear functional Lipschitz in total variation

Let $\Omega$ be a Borel space and let $\mathcal P(\Omega)$ be the space of all Borel probability measures on $\Omega$ endowed with the topology of weak convergence. Define the total variation metric ...
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76 views

Weakly closed convex set and norm convergent

Let $X$ be a normed space and let $\{x_n\}$ be a sequence in $X$ such that $x_n\to x$ weakly. Show that there is a sequence $\{y_n\}$ such that $y_n\in \operatorname{co}\{x_1,...,x_n\}$ and ...
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84 views

Ultra weakly continuous trace on a von Neumann Algebra

Let $M$ be a infinite dimensional von Neumann Algebra with a positive, faithful, ultra weakly continuous trace $tr:M\rightarrow \mathbb{C}$. Is it possible to show that $tr$ is strongly continuous?
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29 views

generalized expression required

suppose i have a set $ {0,1,2.......x-1}$ Now I am generating an i length sequence using the numbers from above set...${a0,a1,....ai}$ where all $ai$$>=0 $ and $ai<=x-1$ Note numbers may ...
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74 views

Functional Analysis Question

Let $(X,\|\cdot\|_1)$ and $(X,\|\cdot\|_1)$ be Banach spaces. Does it imply that $\|\cdot\|_1-\|\cdot\|_2$ (equivalent)? It is know that if $\|\cdot\|_1-\|\cdot\|_2$ and $(X,\|\cdot\|_1)$, then ...
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32 views

Functional characterization of zeroth law of thermodynamics [Sepration of Variables]

Zeroth law of thermodynamics is stated also as: If A is in thermal equilibrium with B and if B is in thermal equilibrium with C, then A is in thermal equilibrium with C. This can be formulated ...
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37 views

Differentiate a log of $L^p$ norm, don't understand this result

I'm reading this paper. In it, the authors show this lemma: And then they prove this lemma My question is: I have no idea how they get the result in Lemma 3.2. Do we not get $$\frac{d}{ds}\log ...
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94 views

A locally convex space is metrizable if and only if it is first countable

I'm studying Functional Analysis by myself. the following is an exercise while I'm not sure about my answer. If $X$ is a locally convex space (LCS), show that $X$ is metrizable if and only if $X$ is ...