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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When does an integral operator belong to the Schatten - von Neumann class in terms of its kernel?

It is well known that an integral operator $X: L^2(\mu)\to L^2(\nu)$ with kernel $k(x, y)$ belongs to the Schatten -- von Neumann class $\mathfrak S_2$ if and only if $\int |k(x, y)|^2\, d\mu(x)\, ...
4
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1answer
80 views

Proof that $(L^1)\neq(L^\infty)^\ast$

I have seen a "proof" that $L^1\neq(L^\infty)^\ast$ which goes as follows: show that there is an element of $(L^\infty)^\ast$ which is not in the image of the canonical map ...
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2answers
52 views

continuity in the strong topology implies continuity in the weak one

I have to prove that if $T:(E,\|\cdot\|_E)\rightarrow (F,\|\cdot\|_F)$ is a continuous and linear operator, and $x_h\rightharpoonup x$ in $E$, than $Tx_h\rightharpoonup Tx$ in $F$. So we know that $T$ ...
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1answer
33 views

What is the domain of an operator?

There seems to be a lot confusion on this notion of a domain of an operator $D(A)$ where $A$ is an operator. Can someone use a simple example to illustrate exactly what this is? Say, let $A$ be a ...
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28 views

Domain Schrödinger equation

If I have a self-adjoint (TIME INDEPENDENT) operator $H: D(H) \subset L^2([0,x]) \rightarrow L^2([0,x])$ with and I want to define the appropriate domain for $$ (Hf)(x,t) = i \partial_t f(x,t),$$ ...
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43 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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57 views

Proper domain for Laplacian

it is well known that the spherical harmonics are eigenfunctions to the 3D Laplacian(angular part). But my question is: What is the right domain for this operator so that we actually get these ...
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26 views

Extension of function with values in a Banach space

I want to prove the following Let $E,X$ be Banach spaces, and $Y\subset E$ a closed subspace with codimension $1$. Let $T:Y \to X$ be a continuous linear function. Then there exists a continuous ...
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1answer
58 views

Is the space of continuous and bijective functions $f\colon [0,1] \to [0,1]$ complete?

Let $X$ be the space of continuous and bijective functions $f$, such that $$ f\colon [0,1] \to [0,1] \quad , \quad f(0)=0 \quad , \quad f(1)=1 \, .$$ Is $X$ complete (under the supremum norm $ ...
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46 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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21 views

Riesz (Hilbert-space) representation theorem and dirac delta on $\mathcal{C}_{0}$

I am thinking about this for a while now, but don't get near an understanding, so I must have gotten something important wrong. I look at $\mathcal{C}_{0}$, the space of countinuous (bounded) ...
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1answer
37 views

Relation between $\epsilon$-pseudospectrum of operators

If $H$ is a Hilbert space and $\sigma_{\epsilon}(T)$ denotes the space of all $\epsilon$-pseudospectrum of the operator $T$ and $S, T\in B(H)$ be such that $TS=ST=0$, why ...
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37 views

Show that $C^1([0,1])$ is not reflexive

Aim of this exercise is proving that $(C^1([0,1]),\|\cdot\|_{C^1})$ is not reflexive. We know that, if $(f_h)_h\subset C^1([0,1])$ is a sequence that weakly converges to $f\in C^1([0,1])$ (that is ...
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1answer
62 views

estimation of gradient

$$(\mathcal{P}_{\varepsilon}) : \left\{\begin{array}{ll} \displaystyle -div\left(A(x)\nabla u_\varepsilon(x)\right)= \dfrac{a(x)}{|u_\varepsilon(x)|+\varepsilon} &\mbox{ in }\Omega \\\\ ...
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1answer
31 views

Fourier transform dependent upon a parameter and $L^2$ convergence

Suppose I know the Fourier transform of a function depending upon a parameter, call it $f_\epsilon(x)$, and that I want to know the Fourier transform of a function $f(x)$. Furthermore, suppose I know ...
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1answer
28 views

Prove that $\lambda_1 x_1 + \lambda_2 x_2 \in S$

Let $S \subseteq V$ be a absolutely convex set, where $V$ is a vector space over $K$. Then show that $\forall x_1, x_2 \in S$ and $\lambda_1, \lambda_2 \in K$ such that $|\lambda_1|+ |\lambda_2| ...
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1answer
22 views

If $A,B,C\in L(X)$ commutes, then $A\leq B,$ and $C\geq 0$ follows $AC\leq BC.$

I do not very well understand functional analysis but this discipline is seems interesting, so some examples of post on this site with the hope that many participants in this site are willing to help ...
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1answer
44 views

There does not exist $f\in (l^\infty)^*$ with $\ker f = c_0$

There does not exist $f\in (l^\infty)^*$ with $\ker f = c_0$. $c_0$ is the closed subspace of $l^\infty$ with the property that if $x = (x_1, x_2,...) \in c_0$ then $$\lim_{n\rightarrow \infty }x_n = ...
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1answer
24 views

$\| f' g\|_{L^2(\mathbb R)} + \|fg'\|_{L^2(\mathbb R)}\le C\left( \|f g\|_{L^2} +\|f'g'\|_{L^2} + \|f'' g \|_{L^2} + \| f g''\|_{L^2}\right)$ holds?

I want to know that the following inequality holds $$ \| f' g\|_{L^2(\mathbb R)} + \|fg'\|_{L^2(\mathbb R)}\le ^\exists C_{>0} \left( \|f g\|_{L^2} +\|f'g'\|_{L^2} + \|f'' g \|_{L^2} + \| f ...
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2answers
26 views

Topological isomorphism vs isometric isomorphism

We say that: $T:(X,\|\cdot\|_X)\rightarrow (Y,\|\cdot\|_Y)$ is a isometric isomorphism if it is a linear isomorphism, and it is an isometry, that is $\|T(x)\|_Y=\|x\|_X\quad \forall x\in X;$ ...
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27 views

give necessary and sufficient conditions that every functional in $w^*-cl M$ be the $w^*$- limit of a sequence from M

Let $X$ be a separable Banach space. If $M$ is a linear manifold in $X^*$ give necessary and sufficient conditions that every functional in $w^*-\mathrm{cl} M$ be the $w^*$- limit of a sequence from ...
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43 views

Completation of an n.v.s. and dimensions of subspaces.

I don't know if the following statement is true: Let $X$ be an n.v.s. with $\text{dim}(X)=\infty$ and not Banach; and $\bar X $ its completation in the bidual space. Let $Y$ be a closed subspace ...
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1answer
42 views

Separability of the space of bounded uniformly continuous functions

Let $(X,\rho)$ a metric space. Do the space $U_b(X)$ of uniformly continuous and bounded real-valued functions on $X$ is separable? It seems that the point is to pass through the Stone-Cech ...
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1answer
32 views

How do the existence and $L^p$ integrability of weak derivative affect the smoothness of Sobolev functions?

In the definition of Sobolev spaces, let us say the space $W^{1,p}(\Omega)$, where $\Omega$ is an open subset of $\mathbb{R}^n$. What contributes more to the smoothness of a function $f$: 1-The fact ...
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0answers
28 views

Relation between $L^1(T)$ and $L^1[0,1]$

I know the question may be too general, but I need to know if there is a way in which I could relate the spaces $L^1(T)$ (where $T=\{e^{2 \pi i x}: x \in [0,1]\}$ and we use the Lebesgue measure on ...
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40 views

local inverse of polynomial

Is there a possibility to invert a polynomial locally? I've got the following problem, concerning control theory: Imagine a ideal amplifier with a feedback loop: Let firstly A be not dependent on ...
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1answer
59 views

How do you show $\int \limits_{X \times Y} f(x,y)\, d\lambda < \infty$ if $\int \limits_{X} \int \limits_{Y} f(x,y) \,d\nu \,d\mu < \infty$?

Suppose $f: X\times Y \rightarrow [0,\infty]$ is a measurable function with respect to the product measure $\lambda$ ( $(X, \Sigma, \mu)$ and $(Y, \tau, \nu)$ are complete measure spaces). Suppose ...
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1answer
41 views

If two functionals have the same kernel, then one is a multiple of the other [duplicate]

I would like some help with this exercise. Suppose that $f_1,\ f_2 \in V^*$ and that $\text{Ker} f_1 = \text{Ker} f_2$. Show that $f_1 = k f_2$ for some scalar $k$. I expect your suggestions. ...
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27 views

Research papers of monotone/pseudomonotone operators with applications to PDEs

I have recently been studying how coercive, pseudomonotone operators are used to prove the existence of solutions to elliptic boundary value problems. I have been studying the book "Nonlinear Partial ...
2
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1answer
38 views

Limit of the p-norm of a function on subdomains equals the p-norm of the function on the union domain

Let $\Omega$ be an open subset in $\mathbb{R}^n$. Given a measurable function $f$, define $$ ||f||_{p,\Omega}=\inf_{a\in\mathbb{R}}||f-a||_{L^p{(\Omega})}. $$ Let $\{\Omega_n\}$ be a sequence of open ...
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0answers
42 views

Taking limit and $\sup$

I was reading a proof for $l^\infty(\mathbb{N})$ is complete. One of the steps is that given $$||x_m - x_n||_\infty = \sup_{i\in \mathbb{N}} | {x_m}_i - {x_n}_i| \leq \epsilon,$$ then for each $I\in ...
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1answer
39 views

$A,B\in L(X)$ is positive semidefinition hermitian operators and $A^2=B^2$, then $A=B.$

Please help demonstrate that applies: If $A,B\in L(X)$ is positive semidefinition hermitian operators and $A^2=B^2$, then $A=B.$ where $X$ is Hilbert space (real or complex), and $L(X$) algebra of ...
4
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1answer
59 views

Convergence in $C(X)$ is uniform convergence.

I read this the convergence in $C(X)$ is uniform convergence. Where $X$ is compact hausdorff topological space and $$C(X)=\{f:X\to\mathbb{C}\;\mid \; f\ \text{is continuous}\}$$ And ...
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1answer
32 views

Prove this inclusion: $\bigcup_{k<p}\ell^k\subsetneq\ell^p$

Let $1<p<\infty$. I have to prove that $$ \bigcup_{k<p}\ell^k\subsetneq\ell^p. $$ I am not able to find a counterexample to prove the inequality.
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38 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 ...
2
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0answers
25 views

difference between uniformly convex norms and strictly subadditive norms?

What is the difference between uniformly convex norms and strictly subadditive norms? why we need to define two above concept? how they help us to study Banach spaces? Is the norm induced by ...
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1answer
21 views

convex weak* sequentially closed subset of a separable Banach space implies weak* closed

I'm studying Conway's a course in Functional Analysis by myself. The following is corollary 6.12.7 of this book. If $X$ is a separable Banach space and $A$ is a convex subset of $X^*$ that is weak* ...
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47 views

Positive part of $y$ with $y\in L^2(0,T; H_0^1(\Omega))$ and $y'\in L^2(0,T; H^{-1}(\Omega))$

Let $\Omega \subset \mathbb R^n$ be a domain, sufficiently smooth. Let $T>0$. Define the space $W(0,T)$ by $$ W(0,T) = \{ y \in L^2(0,T; H^1_0(\Omega)): \ y'\in L^2(0,T;H^{-1}(\Omega)),\ $$ where ...
3
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1answer
39 views

Rellich's theorem for Sobolev space on the torus

From John Roe: Elliptic operators, topology and asymptotic methods, page 73: Let $H^{k}$ be the Soblev space defined on the torus $\mathbb{T}^{n}$ with the discrete $k$-norm: $$ \langle f_{1}, ...
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1answer
64 views

Formulas for Schrödinger unitary groups of operators

Let $\Omega$ an open set of $\mathbb{R}^n$. Consider the Hilbert space $X=L^{2}\left(\Omega\right)$ and the Schrödinger operator $A=i\Delta$ defined on the domain $D(A)=H^2(\Omega)$. Is there any ...
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2answers
24 views

Space of bounded functions vs. bounded space of functions.

Suppose I have a bounded set of functions, say $B\subset C[0,1]$. What exactly does this mean? I.e. is a bounded set of continuous functions equivalent to a set of continuous bounded functions? For ...
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1answer
44 views

$L_H(X)$ is real vector space,

Please help demonstrate that applies: $L_H(X)$ is real vector space, where $X$ is Hilbert space (real or complex), and $L_H(X)$ the set of all hermitian operator on $L(X).$ Thanks for your help and ...
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1answer
38 views

Two linear functionals are equal

Let, $f$ and $g$ be two linear functionals such that ker$f$=ker $g$ and $f(a)$=$g(a)$. Then to prove $f(x)$=$g(x)$.
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1answer
56 views

Show $T: C([0,1]) \rightarrow C([0,1])$ is compact

Consider $T: C([0,1]) \rightarrow C([0,1])$ defined by $$(Tf)(t) := \int_0^1 \kappa_t(s)f(s)ds,$$ where $\kappa:[0,1]^2 \rightarrow \mathbb{R}$ satisfies the following properties: for all $t\in ...
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1answer
43 views

How to show $\operatorname{codim}(\operatorname{Ker} f)=1$ if f is linear?

Let L be linear space and $f :L\to \Bbb R(\Bbb C)$ is linear functional. $\DeclareMathOperator{\Ker}{Ker} \DeclareMathOperator{\codim}{codim}$ $\Ker f$ is a linear subspace and $\codim(\Ker f)=1$ ...
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27 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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0answers
30 views

Different Formulations of Riesz' lemma

Version I: Let $U$ be a closed subspace of the normed space $X$ with $U \ne X$. Also let $0 < \delta < 1$, then there exists $x_{\delta} \in X$ with $||x_{\delta}|| = 1$ and $$ || x_{\delta} ...
2
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0answers
40 views

Shorter proof for $T$ compact and $x_n \to x$ weaky then $Tx_n \to Tx$ strongly

I proved that if $X,Y$ are Banach spaces and $T: X \to Y$ is compact and $x_n \to x$ weakly then $Tx_n \to Tx$ strongly. I am now wondering if there is a shorter proof? Here is my proof: Let $x_n ...
0
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1answer
32 views

Essential support vs. classical support for a continuous function

The essential support of a function $f:\Bbb R^n\rightarrow \Bbb R$ is defined in the following way: Let's denote $\mathcal A_f=\{\omega \subset \Bbb R^n: \omega \quad \text{open}, \quad f(x)=0\quad ...
1
vote
1answer
47 views

If holomorphic $\{f_n\}\to f$ uniformly on compact subsets of $U$, then do $f_n$ and $f$ eventually have the same number of zeros?

Let $U$ be an open subset of $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $U$ such that $f_n\to f$ uniformly on any compact subset $K$ in $U$. Suppose $f$ is not constant, ...