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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Theorem 6.28 of Rudin's Functional Analysis

I am trying for some time to prove theorem 6.28 of Functional Analysis by Rudin.The theorem says that if Ω is an open subset of $\mathbb{R}^n$ and $Λ\in D^{'}(Ω)$,then there is a family ...
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50 views

Subsequences and blocks of Schauder bases

Suppose $X$ is a Banach space and $(e_n)$ and $(f_n)$ are both Schauder bases of $X$. Does there exist a proper closed subspace $Y\subset X$, and appropriate subsequences of $(x_n)$ and $(y_n)$ that ...
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37 views

Jordan normal form

Let $H$ a Hilbert space and let $T\in B(H)$ a bounded operator on H, my question is if it exist a theorem about some "decomposition" of type Jordan canonical form in a general Hilbert space, and how ...
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Can we have extension of Mercer theorem to interpolation?

This question is related to Mercer theorem, Reproducible kernel Hilbert space(RKHS) and interpolation. The wikipedia links are https://en.wikipedia.org/wiki/Mercer%27s_theorem and ...
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Find the iverse of the followning bounded operator?

The following definition and Theorem are given in the book "A short course on operator semigroup" by the author "K-J Engel and R Nagel". Sectoral operator: A closed linear operator $(A,D(A))$ in ...
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An analytical expression for the degree of a map from the sphere to itself

Good morning to everyone. I have found in this paper the following statement (not verbatim): "Let $\phi$ be a smooth map from $\mathbb{R}^2$ to the $3$-dimensional sphere $S^2$ which is constant far ...
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Does $\chi_{A_\epsilon} \to \chi_{\{x : f(x) = 0\}}$ if $A_\epsilon = \{ x \in \Omega : 0 \leq f(x) < \epsilon\}$?

Define $A_\epsilon = \{ x \in \Omega : 0 \leq f(x) < \epsilon\}$ where $f$ is a given function say in $L^1(\Omega)$. Is it true that $$\chi_{A_\epsilon} \to \chi_{\{x : f(x) = 0\}}$$ pointwise? ...
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56 views

Quotient spaces - $\Bbb R^1\hookrightarrow \Bbb R^3$

I am trying to understand quotient spaces, and I constructed my own example to do this: $(\Bbb R=\{(a,0,0)|a\in \Bbb R^1\}) \hookrightarrow (\Bbb R^3=\{(\alpha,\beta,\gamma)|\alpha,\beta,\gamma \in ...
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Is there any parameter space of Cramér–Rao_bound

It is known that Cramér–Rao_bound is the lower bound of variance of a parameter. A useful link is https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Rao_bound There is also a term called ...
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51 views

Weak (operator) null sequence is bounded and pointwise convergent to zero

I was reading Diestel book (Absolutely Summing Operators) and it says: "(...) let $(f_n)$ be any weak null sequence in $\mathcal{C}(K)$. Then $(f_n)$ is bounded and converges pointwise to zero." I ...
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Inequality in a Dirichlet BVP

For the Drichlet boundary value problem $Lu = -u''+p(x)u'+q(x)u = f(x), \; \; \; x \in I=[a,b]$. with $u(a)=u(b)=0$. Then for $v \in H^2(I) \cap H_0^1(I)$ show that $\left\lvert \right\rvert v ...
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60 views

Additional assumptions on function to ensure uniform convergence

Given a sequence $u=(u_n)_{n\geq1}$ converging to $1$, I would like to prove uniform convergence of the sequence of functions $f_n$ defined by $f_n(x)=f(u_n x)$ for $x\in\mathbb{R}_+$ to the function ...
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29 views

Is $\mathbb R^N$ an $C$-distinguished topological space?

I am reading a paper which has some complicated construction on a Hausdorff topological space called $C$-distinguished topological space. The paper says that a $C$-distinguished topological space $X$ ...
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30 views

Generate complete functions set in hilbert space

It is just a curiosity, but is there a general method (or a class of methods) that allows to derive orthonormal complete function set for a given hilbert space? (Except Gram Shmidt algorithm and ...
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27 views

How to solve differential equations for linear operators?

I want to solve the differential equation $$ BA = \frac{\partial}{\partial t} A $$ for $A$. Here $A : H_1 \mapsto H_2$ and $B : H_2 \mapsto H_2$ are operators and $H_1, H_2$ are some Hilbert spaces. ...
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26 views

Abelian Algebras: Generator

Given a Hilbert space $\mathcal{H}$. Consider normals: $$N:\mathcal{D}(N)\subseteq\mathcal{H}\to\mathcal{H}:\quad N^*N=NN^*$$ Denote their algebra: ...
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49 views

Example of a well defined functional integral?

So I was playing around with the notion of a functional integral. Basically given a set $S$ of functions we can define $$ \int_{f \in S} L(f) $$ As the sum of of every function $f$ evaluated by ...
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A doubt regarding derivative of convolution!!

In the following calculation: $\int_{\mathbb R^{d}} u_{o \epsilon} div (\phi) dx = \int_{\mathbb R^{d}} (u_{o} * \psi_{\epsilon}) div(\phi) dx = \sum_{i=1}^{d} \int_{\mathbb R^{d}} ( u_{o} * ...
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33 views

Gaussian unitary dilation of Gaussian channels

I am starting with a few definitions. All these are standard and can be accessed from some quantum information or quantum physics books, for instance the books by Holevo or Parthasarathy. The question ...
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37 views

Direct Integral: Dimension

Direct Integral Given a Borel space $\Omega$ with measure $\mu$. Given Hilbert spaces $\mathcal{h}_x$ for $x\in\Omega$; set $\mathcal{h}:=\bigcup_{x\in\Omega}\mathcal{h}_x$. Regard the function ...
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Consider $c_{00}$ as a subspace of $(\ell^p,\|\cdot\|_p)$. Show that the closure of $(c_{00},\|\cdot\|_1)$ is $\ell^1$

Consider $c_{00}$ as a subspace of $(\ell^p,\|\cdot\|_p)$. Show that the closure of $(c_{00},\|\cdot\|_1)$ is $\ell^1$, closure of $(c_{00},\|\cdot\|_2)$ is $\ell^2$ and closure of ...
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33 views

Is this function uniformly convex?

I am using the following definition of uniformly convex: A continuous functional $G:Y \to R$ is uniformly convex on a ball $$B(0,\delta):=\{y\in Y : \|y\| < \delta\}$$ if there exists a ...
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78 views

The space of continuous functions as a dual space

Let $X$ be some topological Hausdorff space and $C_b(X)$ the space of bounded complex continuous functions on $X$. Is there a Banach space $B$ such that $B^* \simeq C_b (X)$? I know of a very similar ...
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89 views

Alternative ways to prove $\{f:f(0)=\sum_k f(\frac{k}{\sqrt{n}})g_n (k)\}$ is dense in $\{f\in C^2 (\mathbb{R}) : f(0)=\int_{\mathbb{R}} f(u)g(u)du\}$

I want to prove that $$E:=\bigcap_{n\geq 1} \left\{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f\left(\frac{k}{\sqrt{n}}\right)g_n (k)\right\}$$ is a dense subset of: $$F:=\left\{f\in C^2 (\mathbb{R}) ...
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59 views

Operator form $L^2$ space to$L^1$

Can we have an operator such that it transforms an element of $L^2$ to $L^1$? Is this a valid question or this is incorrect? We can consider the measure space as finite.
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25 views

Weak harnack type inequality

I have reached a lemma which I do not have any reference and hint for it. Assume $ \Omega \subset \mathbb{R^N} $ is smooth bounded domain and let $u$ be a positive distributional supersolution ...
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23 views

The relation of the Homogeneous Sobolev norm and general Sobolev norm

I'm wondering if the inequality $$ \left\| F\right\|_{\dot H^k(\mathbb R^n)} \le C\left\| f\right\|_{L^\infty(\mathbb R^n)} \left\| f\right\|_{\dot H^k(\mathbb R^n)} $$ holds for $k\in[0,10]$ then $$ ...
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38 views

topology of uniform convergence on compacts and strong operator topology

I am trying to understand the proof for some lemma in a book, but the part the authors label as trivial is not trivial to me at all. Any help is appreciated. Here is the trivial part of the lemma: ...
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24 views

Is $\{{\operatorname{sinc}}\big({z}-n\big)\}_{n\in \mathbb Z}$ a Riesz basis for $PW_\pi$?

N.K. Bari has proved (see N.K. Bari, "Biorthogonal systems and bases in Hilbert space" Uchen. Zap. Moskov. Gos. Univ. , $148$ : $4$ ($1951$) pp. $69–107$) that a system $\{\phi_n\}$ is a Riesz system ...
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49 views

Special case for Riesz Representation Theorem

The usual Riesz Representation theorem can be found at page 49 in this book. Here I modify it a little bit and providing the following version Riesz representation theorem, where I replaced $\mathbb ...
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23 views

Multiplication by a Cutoff and Convergence in $H^s(\mathbb R^n)$

I'm trying to teach myself some things about Sobolev spaces out of McLean, Strongly Elliptic Systems and Boundary Integral Equations. Exercise 3.14 has me stumped for no reason: Let $K_j \subset ...
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34 views

Well-posedness of nonlinear PDE system

The surface is parametrized by two variables $\sigma_1$ and $\sigma_2$. Moreover, this surface evolves in time. As a result, coordinates of the surface are: $\vec{F} =[x(\sigma_1,\sigma_2 , t), ...
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54 views

Prob. 6, Sec. 4.3 in Kreyszig's Functional Analysis Book: What are all possible extensions?

Here's Theorem 4.3-2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $X$ be a normed space, let $Z$ be a subspace of $X$, and let $f$ be a bounded linear functional ...
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44 views

Compact embedding of Banach spaces; when does a bounded sequence converge in the bigger space?

Let $X$ and $Y$ be Banach spaces with $X$ compactly embedded in $Y$. What more assumptions do I need on the spaces to ensure that: every sequence $x_n \in X$ which is bounded ($\lVert x_n \rVert_X ...
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23 views

Is the Zariski topology equipped with Eisenstein's metric an analytic submanifold?

Using $M=(C(\mathbb{R}),T_z)$ with the norm $(x,y) \to \log(\partial_x+\partial_y)$, we can easily define a derivative using distributions. I was wondering: Does this make $M$ an analytic manifold ...
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70 views

$L^p$ $L^q$ $L^{\infty}$ inclusion + Folland

Proposition 6.10 in Folland's Analysis book states: If $0 < p < q < r \leq \infty$, then $L^p \cap L^r \subset L^q$ and $\|f\|_q \leq \|f\|_p^\lambda\|f\|_r^{1 - \lambda}$, where ...
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35 views

L1 and L2 regularization and L1 and L2 space

I am looking to characterize the difference of the function obtained in the solution process of $L^1$ and $L^2$ regularization. It is known that $L^1$ regularization gives sparse solutions. In $L^2$ ...
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43 views

Importance of boundedness of classical operators of harmonic analysis

Boundedness of classical operators of harmonic analysis, such as maximal functions, fractional integrals and singular integrals have been extensively investigated in various function spaces. For ...
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70 views

On the square root function of matrices

Let $A, B$ be positive definite matrices and let $P$ be an orthogonal projection. If $A \leq PBP,$ does it follow that $$ A^{1/2} \leq PB^{1/2}P?$$
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26 views

How to extend formula for residue to functional calculus of operators

Suppose $\{X_t\}$ is a stochastic process with the covariance operator $\Gamma$ and the first $d$ eigen values are $\lambda_1\geq\lambda_2\geq \ldots \geq\lambda_d$ with eigen vectors ...
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26 views

Reference For a PDE text that treats non homogenuous boundary conditions Rigorously

I am interested in reading a text or paper where elliptic and parabolic PDE's are discussed on bounded domains with non-homogeneous boundary conditions. I haven't been able to find anything in the ...
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36 views

Stochastic process, Fourier transform and $L^2$

Consider a time domain signal received at a sensor $x(t)$ over some time $t$, and we have performed Fourier transform on $x(t)$ to obtain $X(w)$. While performing Fourier Transform to find the ...
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24 views

A question about passing to limits in Banach lattices

I would be grateful if one could confirm that the following argumentation is fine. Suppose that $L$ is a Banach lattice and $(L_n)$ is an increasing sequence of sublattices of $L$. Given two positive ...
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51 views

Are all orthogonal projections conditional expectations?

When will orthogonal projections coincide with conditional expectations? Does that have something to do with the fact that not all closed subspace are probability spaces? Is it why when we fix a ...
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44 views

An example of frame operator.

A sequence of distinct vectors $\{f_1,f_2,...\}$ belonging to a separable Hilbert space $H$ is said to be a Frame if there exist positive contants $A$, $B$ such that, for $A<B$ and for all $f\in ...
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36 views

series of linear operators

Let $\mathcal{B}(\mathcal{H})$ be the Banach space of bounded linear operators on a complex, separable, infinite-dimensional Hilbert space $\mathcal{H}$. It is well known that ...
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56 views

Is it possible to approximate $cos(x)$ with a linear combination of Gaussians $e^{-x^2}$?

I am interested in approximating $\cos x$ with a linear combination of $e^{-x^2}$. I am not an expert in approximation theory but there are a couple things that give me a bit of hope that it might be ...
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43 views

Dual of finite dimensional Hilbert space.

The dual space H* is the space of all continuous linear functions from the space H into the base field. It carries a natural norm, defined by $$\|\varphi\| = \sup_{\|x\|=1, x\in H} |\varphi(x)|.$$ The ...
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51 views

Question about weak derivatives

I have a question about weak derivatives. Let $u,v \in L^{1}_{loc}(U)$ (the space of locally integrable functions) for some open set $\emptyset \neq U \in \mathbb{R}^{n}$ We often say that $v$ is the ...
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26 views

Question about sub differentiability of convex function

I am reading this book to study sub differentiability. On page 20 it says, as $V$ is a normed linear space for simplification, "a continuous affine function $l(v)$: $V\to \mathbb R$ everywhere less ...