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Questions tagged [functional-analysis]

Functional analysis, 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 and other topics. For basic ...

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Infinite orthormal set in an inner product space is closed, bounded and non-compact in the metric space with the induced metric by the inner product

Assume an inner product space $V$ has an infinite orthonormal set $S$. Let $d$ be the metric induced by the inner product. Show that $S$ is closed, bounded and not compact in the metric space $(V,d)...
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Probably of winning with 2 dice (maximum of them) against another one

could some of you help me to find out what is the probability of A) obtain with two dice a greather number than another die? B) and if the dice are 3 how can I do? Not the sum of the 2 dice, but the ...
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About this space subset recording form

$\{\overline{v}\in\mathbb{R}^n \mid \sum_{i = 0}^{n} x_i = 1, x_i\in [0,1]\} = \triangle_{n-1}$ If I will change some parameters, I can have: $\{\overline{v}\in\mathbb{R}^n \mid \sum_{i = 0}^{n} x_i =...
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Classification of $C^*$ algebras whose subalgebra generated by projections is a von neumann algebra

Inspired by this question we ask the following question: Is there a complete classification of all unital $C^*$ algebra $A$ for which the following subalgebra $B$ is a von Neumann algebra? Is ...
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Raise as a problem of $ PL $ the following non-linear programming problem $ (P) $ without restrictions:

Be $ f_{i}: \mathbb{R}^{n} \rightarrow \mathbb{R} $, $ m $ related functions of the form: $$ \left \{\begin{array}{l} f_{ i} (x) = a_{i}^{T} x + b_{i} \\ i = 1,2 \cdots, m \end{array} \right. $$ where ...
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14 views

Problem related with the spectrum of a normal operator

I am working on this problem: Let $A\in L(H)$ be a normal operator on a nonzero complex Hilbert space $H$. Prove that $\sigma(A)\cap i\mathbb{R}=\emptyset\iff A+A^* \mathrm{\ is\ bijective}$ I have ...
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Simple proof that an operator is compact

Let $\phi$ be a compactly supported smooth function on $\mathbb{R}$. I'm looking for a simple proof that the operator $$\left(-\frac{d^2}{dx^2}+x^2\right)^{-1}\phi$$ (where $x$ denotes multiplication ...
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Prove that $V=\{(x_1,x_2,x_3,..):(x_1,2x_2,3x_3,..)\in l_\infty\}$ is linear space

We have $V=\{(x_1,x_2,x_3,..):(x_1,2x_2,3x_3,..)\in l_\infty\}$. How to prove that is it linear space? I think, that here maybe we can begin from that $V\subset c_0$...
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Weak convergence improved by Morrey embedding

Let $u_n: [0,T]\times \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be a sequence with \begin{equation} u_n \rightharpoonup u \ \ \ \text{weakly star in } L^2(0,T;W^{1,\infty}(\mathbb{R}^3)) \end{equation} ...
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construct a sequence of operators [on hold]

Let $(H_n)$ be a sequence of different finite dimensional complex Hilbert spaces, $A_n \in B(H_n),tr(A_n) \to 0(n \to \infty)$,but the norm of $A_n$ does not converge to 0,where $tr()$ is the standard ...
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Choosing the Test Function Space when Deriving Weak Form of PDE

I'm trying to understand how to choose the space of test functions when deriving the weak form of a PDE. For my problem specifically, I have one Neumann boundary condition and one Dirichlet boundary ...
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Comprehensive treatment of Functional Analysis

I don't intend to buy many books. Therefore, I am asking for a book containing a comprehensive treatment of functional analysis. Otherwise Kreyszig or Rudin.
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Does a closed right ideal of a C$^*$-algebra have a C$^*$-algebra?

$A$ is an infinite dimensional C$^*$-algebra and $J\subset A$ is a closed right ideal. $A$ and $J$ are infinite dimensional(as a vector space). I want to find an infinite dimensional C$^*$-algebra ...
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Nonlinear Operator compact but not strongly continuous

can someone help me on this problem? Consider $V=L^2(0,1)$ and $g \in V \setminus ${0}$ $ and let $A: V \rightarrow V^\ast$, $v\mapsto Av := g ||v||²$ be an nonlinear Operator. Prove that A is ...
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unbounded differential operator in Lp

By semigroup theory the following are well known Suppose that $A:X \longrightarrow X $ is a bounded (or unbounded) operator on some Banach's space $X$, and given the Cauchy problem: $$\frac{d}{dt}u(t)...
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Proof that for a strongly continuous contraction resolvent, there is exactly one linear operator that generates the resolvent.

I have questions about the proof of the following Proposition from the book Introduction to the Theory of Non-Symmetric Dirichlet forms. First, how do we get the independence of $G_\alpha (B)$ of $\...
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Relationship between kernel function and basis

Is there any relationship between a kernel function and the concept of basis in a vector space? For example does an equality like this $$ g(x) = \int_{\Omega} \alpha(y) K(x,y)dy $$ have anything to ...
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1answer
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Is the set $\overline{\langle \{ f^{*}_{1}f_2: f_1,f_2\in J\}\rangle } $ equal to $B(\ell^2)$?

If $J=\left\lbrace f\in B(\ell^2): f^*(e_1)=0 \right\} $, is the set $\overline{\langle \{ f^{*}_{1}f_2: f_1,f_2\in J\}\rangle } $ equal to $B(\ell^2)$? ( $\ell ^2 $ is the Hilbert space $(\ell^2, \...
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Rudin functional analysis, theorem 2.11 (Open mapping theorem)

Going through the proof of such theorem, there are few bits I don't understand. I'll write down the proof and in in the middle I'll add some comments. Statement: Suppose (a) X is an F-space, ...
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Weak* Convergence exercise

I'm dealing with this exercise about weak* convergence and I'm literally getting lost with indexes. I have this: Let $X := c_0(\mathbb{N}), \hspace{3mm}x_0 \in X^*=\ell^1(\mathbb{N}), \hspace{3mm}\{...
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1answer
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A Hahn-Banach separation theorem argument, claryfying the details

I have a question regarding the proof of proposition 6.1. in https://arxiv.org/pdf/1509.01870.pdf, how exactly Hahn-Banach separation theorem has been used. Proposition 6.1: A discrete group is $C^*$-...
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1answer
24 views

Integral Inequality with L-2 Norm

On page 135 of The Mathematical Theory of Finite Element Methods (Brenner and Scott), I encountered the following inequality: $\left | \int_{\Gamma} \overline{v} - v \, ds \right | \leq |\Gamma |^{...
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1answer
31 views

About normal operator

I am working on a problem regarding the spectrum of a normal operator and get stuck here. Let $H$ be a Hilbert space and $A\in L(H)$ is normal, if we know $\mathrm{Re}\left<x,Ax\right><0$ ...
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1answer
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Two Baire measure, $\left\{A\subseteq X: A \text{ Baire set}\ \mu(A)=\nu(A)\right\}$ is $\sigma$-algebra

pag 235 Barry Simon. A comprehensirve course in analysis Let $X$ compact hausdorff space. Let $\mu,\nu$ baire measures, and $\mu(X)=\nu(X)=1.$ Let $S:=\left\{A\subset X: A \text{ baire set and } \mu(...
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$\{u_m\}$ is bounded in $W^{1, p}(U)$, but does not possess a (norm-)convergent sequence in $L^{p^∗}(U)$

Let $U=B_1(0)$ be the unit ball in $\Bbb{R}^n$, $1≤p<n$, $p^∗=\frac{np}{n−p}$. Consider the sequence: \begin{equation} \label{eq:aqui-le-mostramos-como-hacerle-la-llave-grande} u_m = \...
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1answer
20 views

Minimizing an Integral related to Fourier Series

Consider the inner product space $ C[0,2\pi] $ (or its completion, the Hilbert Space $ L^2[0,2\pi] $) as in 3.5.1 Find the values of $ c_1, c_2, $ and $ c_3 $ which minimize the value of $$ \int_0^{2\...
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Question about $C_c^{\infty}(]-1,1[)$ and Lipschitz spaces

We denote by $D$ the space of the functions $C^{\infty}(]-1,1[)$ and compact support and $F$ the space of the Lipschitzian functions on $[-1,1]$. For $f\in F$ and $\varphi\in D$, we note $B(f,\...
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1answer
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Conditional expectation for a partition

Consider a probability space $(\Omega,\mathcal{F},P)$ and a bounded random variable $X$. Let $(A_n)_{n\ge 1}$ be a countable partition of $\Omega$ and define $\mathcal{A}:=\sigma(\{A_n:n\ge 1\})$. ...
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Does this space enjoy Schur property? (proofcheck)

I need to know if the following reasoning is correct, if there is someone so kind to check it. Let $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$ be the d-dimensional thorus and consider the inhomogeneous ...
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Lipschitzness of derivatives

Let $f \in C^\infty_b(\mathbb{R}^d; \mathbb{R}^d)$, so bounded, infinitely differentiable with bounded derivatives mapping $\mathbb{R}^d$ to $\mathbb{R}^d$. I'll write $|\cdot|$ for the norm on $\...
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Embedding of two normed vector spaces

Let $V,U$ be two normed vector spaces such that $V \subset U$. If people say $i$ is an embedding of $V \to U$, do they mean that $i$ is the identity map which is well-defined since $V \subset U$?
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Short-time Fourier transform of $f\ast g (y)- f\ast g(x)$?

Let $f, g\in \mathcal{S}(\mathbb R)$. Then the convolution of two Schwartz class function is again Schwartz class function, that is, $f\ast g \in \mathcal{S}(\mathbb R).$ It is also known that the ...
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When is a function positively homogeneous?

For example, if my function is: f(x) = 3 How do I establish if it's positively homogeneous or not?
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Taylor's theorem for multivariate functions of matrices

Let $f:\mathbb{R}^{m\times n}\to\mathbb{R}$ be $C^3$, then according to Taylor’s formula in Banach spaces and also chapter 4 of Zeidler's functional analysis, we can write the Taylor's formula for $f$ ...
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Lifting of operator under free action

Suppose a Lie group $K$ acts freely, properly and isometrically on a Riemannian manifold $(M,g)$, which is equipped with a $K$-invariant measure. Then $(M/K,\check{g})$ is a Riemannian manifold, where ...
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Study the convergence of the sequence $f_n(x)=\frac{x-n}{x^2}\cdot\chi_{(n,+\infty)}(x)$

For every $n\in\mathbb{N^+}$, let $f_n:(0,+\infty)\to\mathbb{R}$ be as defined: $$f_n(x)=\frac{x-n}{x^2}\cdot\chi_{(n,+\infty)}(x).$$ Study the convergence of the sequence $\{f_n\}_{n\in\mathbb{N^+}}$...
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What's exactly meant if one is saying that the closure of a multivalued operator is the generator of a $C^0$-semigroup?

Let $E$ be $\mathbb R$-Banach space $A$ be a subspace of $E\times E$ and $$\mathcal D(A):=\left\{x\in E\mid\exists y\in E:(x,y)\in A\right\}$$ $(T(t))_{t\ge0}$ be a contractive $C^0$-semigroup on $\...
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Equivalence of summable conditions

Let $X$ be a Banach space and $(f_n)_{n\ge1}$ is a sequence in $X^*$. Show that $\sum_{n\ge1} f_n(x)$ is summable for each x in $X$ if and only if $\sum_{n\ge1} \phi(f_n)$ is summable for each $\phi$ ...
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Increasing property of solution of heat equation

Let $$u_t - \Delta u = f$$ with $u(t,0) = u(t,1) = 0$ and $u(0,x) = u_0(x)$ given, and $f(t,x)$ is also given. This is the heat equation on the interval $(0,1)$. How can I choose $f$ and $u_0$ to ...
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Does the inner product define a function?

Let $\psi: l^2 \to \mathbb{R}$, I can't understand what this stands for $$<\psi, e_j>=\frac{(-1)^j}{j!}\in\mathbb{R}$$ $e_j$ are zero vectors with a $1$ in the $j$-th position. What ...
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Easy way to understand the definition of orbits, Rudin functional analysis.

In the theorem of Banach-Steinhaus the following defintion of orbits is given. Assuming $X,Y$ are topological vector spaces and $\Gamma$ is a collection of linear mappings from $X$ to $Y$ we define ...
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The set of 3-tuples numerical range of Hermitian operators is not convex

I am trying to find a counter example for the following claim. If $A_1, A_2, \dots, A_n$ are Hermitian operators then the set of all $n$-tuples of the form $\left \{ \left( \left < A_1 f, f \...
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1answer
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Are $\lVert .\rVert _2$ and $\lVert . \rVert_{F} $ equivalent for a matrix?

If $A\in mat_{n\times n}(\mathbb C) $ and $\lVert A\rVert _F $ is equal to the Frobenius norm of $A$ and $\lVert A\rVert_2=(\lambda_{\max }(A^*A))^{\frac{1}{2}} $, Are these norms equivalent? I know ...
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Tangents parallel to one another and finding the perpendicular of a vector function

We have $$x(t)= \begin{pmatrix} 1+t \\ t^2-t \\ 1-t^2 \\ \end{pmatrix}$$ Part 1 Are there $2$ points $x(t_1)$, $x(t_2)$, such that the function’s tangent vectors at these points are parallel ...
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How to prove this result somewhat similar to Du Bois-Reymond's Lemma?

Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded connected set with smooth boundary. Suppose also that for each index $i, j = 1, \dots, n$, $f_{ij}$ is a smooth function. If for every $v\in C^...
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32 views

Using Picard-Lindelof to find a solution to $y'(t,y(t))=t+\sin(y(t))$ where $y(2)=1.$

Consider the initial value problem $y'(t,y(t))=t+sin(y(t))$ with initial condition $y(2)=1$. Find the largest interval $\mathcal{I}\subset \mathbb{R}$ containing $t_0=2$ such that the problem has a ...
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1answer
24 views

Uniformly approximating $f\in \mathcal{C}([1,\infty))$ with polynomials where $\underset{x\rightarrow +\infty}{\lim} f(x)=a$

Suppose $f\in \mathcal{C}([1,\infty))$ and $\underset{x\rightarrow +\infty}{\lim} f(x)=a$. Show that $f$ can be uniformly approximated on $[1,\infty)$ by functions of the form $g(x)=p(1/x)$, where $p$ ...
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13 views

Remark 16 in Chapter 2 from Brezis - Functional Analysis, Sobolev Spaces, and Partial Differential Equations.

Let $E,F$ be two Banach Spaces and $A : D(A) \subset E \to F$ be a linear unbounded operator which is densely defined. Now, we would like to define $A^{*}$ as the adjoint of $A$. Let $A^{*} : D(A^{*}) ...
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102 views

Value of operator norm when $\mathcal{T}f(x)=\int^{x}_{0} f(t)dt$

Let $\mathcal{C}$ be the space of continuous functions on $[0,1]$ equipped with the norm $\|f\|=\int^{1}_{0}|f(t)|dt$. Define a linear map $\mathcal{T}:\mathcal{C}\rightarrow \mathcal{C}$ by $$ \...
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8 views

Iteration of a sequence

I am reading a paper on Piecewise Convex Transformations. I got an inequality as : $||P_{\tau} f||_{\infty} \leq {\frac{1}{\alpha}|| f||_{\infty} + C||f||_{1}}$ After n iterations, we obtain : $||P_{...