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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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Concering the proof of the Naimark Duality Theorem (frames, Riesz bases)

Naimark Duality Theorem: Let $(\varphi_{i})_{i \in I} \subset \mathcal H$. Then $(\varphi_{i})_{i \in I}$ is a frame iff $\exists$ a Hilbert space $\mathcal K \supset \mathcal H$ and a Riesz Basis $(\...
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1answer
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Entropy Theory and Conditional Entropy

Peter Walters An Introduction to Ergodic Theory. Chapter 4 Entropy Page 83 & 84 How did they duduce the last formula ?
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18 views

Tensor product of dual spaces and dual space of tensor product

Let $\mathcal{A}$ and $\mathcal{B}$ be infinite-dimensional C*-algebras, and let $\mathcal{A}^*$ and $\mathcal{B}^*$ denote the space of norm-continuous linear functionals on $\mathcal{A}$ and $\...
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1answer
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$\text{Hol}^2(D)$ is a closed subspace of $L^2(D)$.

Let $D$ be the unit disk in $\mathbb{C}$. I want to prove that $\text{Hol}^2(D)$, the subset of $L^2(D)$ which consists of holomorphic functions, is closed. That is, if $(f_n)$ is a sequence of ...
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31 views

Finding an orthonormal basis of $L^2[0,1]$ consisting of elements from $C[0,1]$

Problem: Let $f \in L^1[0,1]$ but $f\notin L^2[0,1]$. Is there an orthonormal basis of $L^2[0,1]$ consisting elements {g_n} in $C[0,1]$ such that $\int g_n f = 0$? To start with, one can show that ...
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28 views

If $f\in L^\infty(0,1;H_0^1)$, then $\int_0^1fds \in H^1_0$?

Suppose that $f\in L^\infty(0,1;H_0^1(\Omega))$. Is it then true that $\int_0^1f(s)\,ds\in H_0^1(\Omega)$? ($\Omega\subset\mathbb R^d$ is a bounded domain, $H_0^1(\Omega)$ is the Sobolev space ...
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1answer
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Continuity of functional in $E\subset C^1([0,1])$

Let $$E:=\{x\in C^1([0,1]):x(0)=x(1)=0\}.$$ Then $(E,\|\cdot\|)$ is a normed space when $$\|x\|=\sup_{t\in[0,1]}\{|x'(t)|\}.$$ Now fix $s\in[0,1]$ and set $f_s:E\rightarrow\mathbb R$ as $f_s(x)=x(s)$. ...
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18 views

Closed Range Condition

Let $A$ be a bounded linear operator on some separable Hilbert space $\mathcal{H}$. We have the following iff condition for $A$ to have closed range: $\exists C>0$ s.t. $\forall\varphi\in\ker(A)^\...
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connected set of sum of upper semi continuous function

Let $C(X)$:space of continuous functions on a compact space. Consider $f$ and $g :C(X)\rightarrow \mathbb{R} -\{-\infty, \infty \}$ are upper semi continuous. suppose for every $T\in C(X)$ set of $...
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1answer
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Bounded metrics beyond the bound

The standard bounded metric (see slide 3 of 8) is defined as $\overline d(x,y)=\min\{1,d(x,y)\}$. For balls of this metric $B_{r,\overline d}(p) = \{x \in M|\overline d(x,p)<r\}$, they're the same ...
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Frame theory reference request

I want to study basic frame theory on my own. I went through a one semester course in Functional Analysis. Which book/Lecture notes would you like to suggest to me?
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Variational lemma and the Schrödinger equation.

Suppose that for each $t\geq 0$ the function $\psi(x,t)$ is of the class $L^2(\mathbb{R^3};\mathbb{C})$. Suppose also that the function $[\partial_t -H]\psi$, where $H$ is the Hamiltonian operator, is ...
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Do the poles of the resolvent of an operator correspond to eigenvectors?

Consider a matrix $M$ with resolvent $$R_M(z)=(z-M)^{-1}$$ The poles $\lambda_i$ of $R_M(z)$ are eigenvalues of $M$, and the eigenvectors $v_i$ of $R_M$ are also eigenvectors of $M$ such that $$...
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What is the space $H^2([0,1])$ inside $L^2([0,1])$?

I understand that for a complex domain $\Omega$, $H^2(\Omega)$ is the closure of the span of $1, z, z^2, \dots $ . However, I am unsure if this space is any different on $\mathbb{R}$.
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Each square integrable harmonic function can be written as the sum of an antianalytic and an analytic, square integrable functions.

I want to prove that each square integrable harmonic function with respect to the standard Gaussian measure: $\gamma ^{{n}}({\mathbb{C}})={\frac {1}{{\sqrt {2\pi }}^{{n}}}}\int _{{\mathbb{C}}}\exp \...
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Energy functional of Poisson equation

Define $\mathcal F(u) = \int_U \frac{1}{2} |\nabla u|^2 - \int_U fu - \int_{\partial U} gu : C^\infty(\bar U) \to \mathbb R$, where $f \in C^\infty(\bar U), g \in C^\infty(\partial U)$. Suppose $u_0 \...
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Are the poles of the resolvent of a Hermitian operator real?

The resolvent of a matrix $M$ is defined as $$R_M(z)=(z-M)^{-1}.$$ The spectrum of $M$ can then be defined as the set of points $\lambda$ for which $R_M(\lambda)$ is singular. Since all the ...
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1answer
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How to prove inequality $\sum_{j=1}^n\vert x_jy_jz_j\vert\le\Bigl( \sum_{j=1}^n\vert x_j\vert^{p_1}\Bigr)^{\frac{1}{p_1}}…$

I'm new in StackExchange and in functional analysis. I have a problem with one exercise: If $p_1, p_2,p_3\gt1$ are real numbers with $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}=1$ is true, n - a ...
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21 views

Direct proof of Closed Graph Theorem (or Bounded Inverse Theorem) from Uniform Boundedness Principle

I'm looking for a direct proof of the Closed Graph Theorem (or Bounded Inverse Theorem) from the Uniform Boundedness Principle. But I can't find one in the literature. I'm hoping there's a nice ...
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Question about a estimative in Hernández's article

On page 1143 of the article by Hernández, Rabello and Henríquez, at https://ac.els-cdn.com/S0022247X0601033X/1-s2.0-S0022247X0601033X-main.pdf?_tid=c4504062-1c37-4ac3-a2a4-6d7041b2f579&acdnat=...
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1answer
27 views

To represent a bilinear operator on finite dimensional space by a matrix

This exercise somehow connects functional analysis to linear algebra. However, I am still not be able to fully understand the insights behind it. Recall the standard inner product on $\mathbb{C}^n$...
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28 views

Reference request for invariant borel measures on Banach spaces

The following result is standard in Banach space theory: The only locally finite, translation invariant, Borel measure on an infinite-dimensional separable Banach space is the trivial measure that ...
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34 views

Weak convergence and weak* convergence

Let $\Omega$ be some measurable space. Suppose $\{x_n\}$ is a sequence bounded in $L^{\infty}((0,T) ; L^2(\Omega;H^1(\mathbb{R}^d)))$ , then we know there exists a subsequence $\{x_n\}$(denoting same) ...
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37 views

Relationship between energy functions in calculus of variations.

I've been implementing lately few algorithms based on energy functions of the form. $$ E = \int_\Omega \mathcal{L}(x,f,f')dx $$ In the above the cost function $E$ defines an actual energy that we ...
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33 views

Does it make sense to use “linear transformation” to analyze effectiveness of a new technique?

I am relative new in statistical analysis, and I have question when I'm trying to measure the performance of new techniques. Imagine we have $m$ machines to improve with new technique, and $n$ tasks ...
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1answer
25 views

Interchanging limit of sequence of functions

Let $\{u_{j}\}_{j\in\mathbb{N}}$ be a bounded sequence in $L^{\infty}(\Omega)$ for a given smooth bounded domain $\Omega \subset \mathbb{R}^{n}$. Assume $u_{j} \to u\in L^{\infty}(\Omega)$ a.e. in $\...
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Extending the Fourier transform on $L^1(\mathbb{R}^n)$ to $L^2(\mathbb{R}^n)$

We define the Fourier transform of $f \in L^1(\mathbb{R}^n)$ with the usual formula $\int e^{-i k \cdot x} f(x) dx$. This does not work for the functions in $L^2(\mathbb{R}^n)$. The defining integral ...
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1answer
25 views

A bounded linear operator that is close to a unitary operator should be invertible.

Suppose that $T,U$ are bounded operators on a Hilbert space $H$, with $g$ unitary. If $\|T-U\| < 1$, then show that $T$ is invertible. Injectivity is easy: Suppose $z \in kerT$, then $\| (T-U)z \|...
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1answer
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Is there a bounded linear function $F: L^{\infty}([-1,1]) \to \mathbb R$?

Is there a bounded linear function $F: L^{\infty}([-1,1]) \to \mathbb R$ such that $F(u) = u(0)$ for u bounded and continuous at $0$? BASIC ANALYSIS: the set of functions bounded and continuous at $0$...
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1answer
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Counterexample the riesz representation theorem

The exercises shows that the Riesz Representation does not hold on infinite-dimensional inner product spaces. I need help. Suppose $C_{\mathbb{R}}([-1, 1])$ is the vector space of continuous real-...
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1answer
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Intersection of kernels which contains a subspace

Let $X$ be a normed space. Assume $Y\subset X$ is a linear subspace of $X$. Prove that $$\overline{Y}=\bigcap_{f\in X^*,\mathrm{ker}f\supset Y}\mathrm{ker}f$$ I was able to prove that $\overline{Y}\...
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Pointwise Convergence in the Space of Sequences

I am asked in a question to show that: In (s) ,the space of sequences of complex numbers, show that convergence and pointwise convergence is equivalent. I am having trouble understanding what ...
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2answers
31 views

The image of a multiplication operator has to be either the whole space or of first Baire Category

Let $\phi \in C[0,1]$ and $T_{\phi}: C[0,1] \to C[0,1]$ to be the multiplication operator such that $T_{\phi}(f) = \phi f$, then either the range of it is the whole space or it is of the first Baire ...
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Example: $F(t)$ is strongly continuous on $X$ but $F^*(t)$ is not

Let $X$ be a reflexive Banach space. I am looking for examples of the strongly continuous operators $F(t):t\mapsto \mathcal{L}(X)$ such their adjoint on $X$ for every $t$ is not a strongly continuous ...
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A Hilbert space is separable if and only if it admits a countable orthonormal basis [duplicate]

According to Wikipedia: A Hilbert space is separable if and only if it admits a countable orthonormal basis While this statement seems very reasonable, it is not clear to me how one would go about ...
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21 views

Manual inverse Fourier transformation in different time stepping.

I just trying to get analytical expression for a discrete signal. I am using matlab fft for this. After transforming I get a series of Y(k) coefficient. Next I am trying to use series of $$X(j) = \...
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2answers
45 views

Let $A,B \in \mathcal{M}_{n\times m}(\mathbb{R})$. Show that $\|A^TB \|_F \leq \|A \|_F \|B \|_{\text{op}}$

Definition: The Operator norm, denoted $\| \cdot \|_{\text{op}}$ is defined to be $$\| B \|_{\text{op}} = \inf\left\{ c \geq 0 : \|Bv \|\leq c\|v \|\ , \forall v \in V\right\}$$ Definition: The ...
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Projection of Shrinked Vector

Assume $\boldsymbol{e}_1$ and $\boldsymbol{e}_2$ are a set of orthonormal basis for $\mathbb{R}^2$. There is a real vector $\boldsymbol{u}$ such that $|\boldsymbol{e}_2^T\boldsymbol{u}|\le C |\...
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37 views

Convergence in $l_p$ space

Let $1<p<\infty$. Also let $\{x_n\}\subset l_p$ such that $\sum_{j=1}^{\infty} x_n(j)y(j)\to 0$ $ \forall y \in l_q$, $\frac{1}{p} + \frac{1}{q} = 1$. Then show that sup$_n ||x_n||_p < \infty$...
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In Hilbert space find the radius of convergence and h0 and norm. [on hold]

question 3 in A course in functional analysis by conway pg 13
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Hessian Metric and Bregman divergence

I read from a paper that Bregman divergence is an approximation to the Hessian metric when the two points are nearby. What is the definition of Hessian metric? How can we derive this approximation?
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32 views

Essentially Smooth Function

A function $f:\mathbb{R}^n \to \mathbb{R}$ is essentially smooth if $f\in \mathcal{C}^1$ and $||\nabla f(x)||_* \to \infty$ as $||x||\to\infty$. What is the intuition of essentially smoothness? ($||.||...
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Can someone provide a 'map' to measure theory, Hilbert spaces, metric spaces, point-set topology, etc?

All analysis books (Rudin, Folland, etc.) cover the same material. But if you are trying to study it on your own,it would be helpful to know (1) what motivated the various developments and (2) how ...
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2answers
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Is my proof that $l^1$ is isometrically isomorphic to $c_0^*$ correct?

This is a classic exercise of functional analysis, but I do not fully understand it after reading many answers in textbooks. So I am trying to reorganize the proof step by step in details. I am hoping ...
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1answer
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Convergence pointwise of limited operators, then the sequence of norms is limited

I'm studying funcional analysis and our professor left the following problem: Let $E$, $F$ be normed spaces, and $\left( A_n\right)$ be a sequence of limited linear operators from E to F, such that, ...
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24 views

Can all real polynomials be written as the sum of periodic functions?

I was reading the article: https://mathblag.wordpress.com/2013/09/01/sums-of-periodic-functions/ and the author makes the claim that If P is a polynomial function of a real variable, and the ...
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1answer
20 views

Proving null space of adjoint operator is equal to the dense range

I'm close to being done with a proof, but I'm not sure whether it's correct and there are a few things I'm unsure about. I'll start off by stating the theorem I'm trying to prove: $\textit{For a ...
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2answers
29 views

Is there a sequence in L1-space whose infinity is zero norm and two norm is infinity? [on hold]

When reading to the book of Markus Haase , I have come across to the question which is "Find an example of a sequence in L1 whose 2-norm is infinity and infinity-norm is zero? " Can anyone be helpful. ...
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1answer
39 views

Best approximation of vector.

Consider the inner product space $P([0,1])$ of all real polynomials on $[0,1]$ with inner product $\langle f,g\rangle=\int_0^1f(x)g(x)dx$ and $V=span\{t^2\}$. Let $h(t) \in V$ be such that $\|(2t-1)-...
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15 views

For which functions/distributions X, $S(\mathbb R) \ast X \subset \ell ^{1}({\mathbb Z})$?

Let $f\in \mathcal{S}(\mathbb R^d)$ (Schwartz space) and define $f_k(x) =f(x-k) \ (k\in \mathbb Z^d)$ (translation of $f$). Question: For which kernel $h:\mathbb R^d \to \mathbb C,$ one can ...