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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A question about the second differential

Hi I have a doubt: What is the matrix associated at the second differential? Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$, differiantable and let $df: \mathbb{R}^n \rightarrow \mathbb{R}$, ...
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1answer
90 views

Spectra of operators

Please help me proof a theorem: If $\mathfrak{U}$ is a complex, commutative Banach algebra with identity and $x\in\mathfrak{U}$, then $$ \sigma(x)=\{\phi(x):\phi \text{ is a homomorphism of } ...
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89 views

Convergence of Schwartz kernels implies convergence of operators

Let $K$ be a smoothing operator on $\mathbb{R}^n$, i.e., it defines a map on all Sobolev spaces $K\colon H^r(\mathbb{R}^n) \to H^s(\mathbb{R}^n)$ for all $r, s \in \mathbb{R}$. Now (a variation of) ...
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1answer
30 views

Uniformizability of a space

Let $E$ be a topological space. For $x \in E$,the nhds of $x$ which are both closed and open form a fundamental system of nhds of $x$.Show that E is uniformizable. Check here for definition of ...
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2answers
347 views

Weak compactness of Sobolev spaces

I am trying to understand a proof in Evan's book "Partial Differential Equations". We have a sequence $(u_n)_{n\in\mathbb{N}}$ in $L^q(U)$ where $U$ is a bounded open set of $\mathbb{R}^m$. We know ...
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96 views

Is $H(f)={1\over x} \int_0^x f(t)dt$ compact? [duplicate]

Possible Duplicate: Operators on $C([0,1])$ that is compact or not. I feel a bit bad about raising this question since similiar question have been answered many times. But I couldn't find ...
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1answer
73 views

Maps sending weakly convergent sequences to weakly convergent sequences are continuous?

Well, the question is in the title. I understand that they are continuous in the weak topology, but can't see that it must hold for the norm topology. Please help me.
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1answer
118 views

Weak compactness

Define a map $\varphi \colon [0,1]\to C[0,1]^*$ by $\varphi(x) = \delta_x$. Then $\varphi$ is a homeomorphism for the w*-topology. Let $K$ denote the image of $\varphi$. I have two questions: 1) Is ...
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1answer
189 views

Spectrum in an separable Hilbert space

Let $H$ be a separable Hilbert space with orthonormal basis $\{e_i\}$. Let $(c_n)$ be a bounded sequence of complex numbers and consider the bounded linear operator $T$ on $H$ defined by $$Tx = ...
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3answers
255 views

The space $C_b(\mathbb{R})$ is complete

Let $C_b(\mathbb{R})$be the space of all bounded continuous functions on $\mathbb{R}$, normed with $$\|f\|= \sup_{x\in \mathbb{R}}|f(x)|$$ Show that this space is complete. Complete mean that all ...
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68 views

Properties of the spectrum

Let $\rho$ denote the resolvent of a closed operator and if $\lambda \in \rho(A)$, define $R(\lambda,A) := (\lambda I -A)^{-1}$. If $\mu$ is sufficiently small ...
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1answer
73 views

About an $L^\infty$ equality

Let $f$ be continuous on $\Bbb R$ and $c $ be a positive real number. Let $x \in [0,K]$ for some $K >0$. Then does there exist a constant $C >0$ such that $$ \| f(cx) \|_{L^\infty([0,K])} = C ...
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2answers
99 views

Exercise: If $\{f(x_n)\}$ is Cauchy $\forall f \in X^\ast$ then $\exists x \in X : x_n \rightarrow x$ weakly

I'm working on this exercise (not homework) and I would gladly welcome some hints for how to solve it! Excercise: $X$ is a reflexive Banach space and $\{x_n \} \in X$. Prove that if $\{f(x_n)\}$ is ...
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3answers
1k views

Paul Erdos's Two-Line Functional Analysis Proof

Legends hold that once upon a time, some mathematicians were rather pleased about a 30-ish page result in functional analysis. Paul Erdos, upon learning of the problem, spent ten or so minutes ...
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1answer
297 views

Compute the spectrum for a operator

Find the spectrum of the operator $$ \begin{split} A & \colon C[0,1] \rightarrow C[0,1] \\ & f \mapsto (Af)(x) := f(x) + \int_0^x f(t)dt \end{split} $$ P.S.: I know the spectrum ...
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0answers
81 views

Topological Space in which every compact subset is metrizable

Is there an (more or less) established name for the class of topological spaces in which every compact subset is metrizable? This is true for example in (LF)-spaces (inductive limits of ...
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0answers
86 views

Banach limit and its commutative counterpart, what do they tell us?

A Banach limit is a continuous linear functional $\Lambda$ on $\ell^{\infty}(\mathbb{N})$ satisfying: $\|\Lambda\|=\Lambda(1,1,1,\cdots)=1$; and ...
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1answer
54 views

Deriving an estimate for $L^\infty$ norm of a function

Let $f \in C^\infty (\Bbb R) $, and $f(0) = 0$. Assume if $|w| \leq \delta$ then $| f(w) | \leq c |w|^a$ for some fixed $a \in \Bbb N$. Now let $\| w \|_{L^\infty(\Bbb R)} \leq \delta$. Then can ...
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1answer
258 views

Application of the Hahn- Banach theorem

Let $E$ be a normed space and $F$ be a subspace of $E$. Show that $F$ is dense in $E$ if and only if all the linear and continuous functional on $E$ satisfying $f\vert _F=0 $ are identically zero ($f ...
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1answer
41 views

Skew-symmetric unitary on $\mathcal{B(H)}$

We know that there exists skew-symmetric unitary on $\mathcal{B(H)}$ when $\mathcal{H}$ is of even dimensions. In particular for $\mathcal{H}=\mathbb{C}^2$, any such matrix is scalar multiple of Pauli ...
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1answer
143 views

Variation of the fundamental lemma of calculus of variation

Let $$C^1_0[a,b]:=\{f \ C^1[a,b]|f(a)=f(b)=0\}.$$ Providing $C^1_0[a,b]$ is dense in $L^2[a,b]$, I want to prove the following statement: if for $g,h\in L^2[a,b]$, $$\int_a^b g \phi \,dx =\int_a^b h ...
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2answers
135 views

Why for a compact metric probability space, any Borel subset can be approximated by compact set?

Let $X$ be a compact metric space with a Probability Borel measure $\mu$. Let $C$ be any Borel subset of $X$. Then for any small positive number $a$, we can find compact set $K$ such that $K$ is ...
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1answer
267 views

Exercise: Application of The Uniform Boundedness Principle

I'm working on this exercise (not homework) and I would gladly welcome some hints for how to solve it! Exercise: Let $1 \leq p,q \leq \infty$ be conjugate exponents. Let $a=(a_1,a_2,...)$ be a ...
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1answer
66 views

Proving the following moment distribution function.

I am trying to prove the following relation , If $u \in L^p(\Omega)$ $\Omega \subset R^n $and $0 < p <\infty$ , the the following relation is valid , $$\|u\|_{L^p(\Omega)}^p = p\int_0^\infty ...
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1answer
540 views

How is it that the derivative operator is a closed linear operator?

By definition, if the derivative operator $D:C^1[-1,1]\to C^1[-1,1]$ is closed, then it should be the case that, given any sequence $\{x_n\}$ in $C^1[-1,1]$, and given that $x_n\to x$ as $n\to\infty$ ...
2
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1answer
177 views

weak derivative and continuous functions (functionals, distributions)

Suppose $\varphi \in C_c^\infty(0,T; H^1(\Omega))$ is a $H^1(\Omega)$-valued test function (it vanishes at $t=0$ and $t= T$), and $f \in C^1(0,T \times \Omega)$. Let $w \in L^2(0,T;H^1(\Omega)$ with ...
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1answer
151 views

Polar decomposition of right shift operator

Let $T: H\to H$ be a bounded operator on Hilbert space $H$. $T(e_n) = a_n e_{n+1}$ where $\{e_n\}$ is orthonormal basis and $\{a_n\}$ is bounded sequence. What is the polar decomposition of $T$? For ...
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63 views

Is $f \in C^1([0,T]\times \Omega) \iff f \in C([0,T]; C^1(\Omega))$?

Is $f \in C^1([0,T]\times \Omega) \iff f \in C([0,T]; C^1(\Omega))$ My guess is no, I think the RHS is a bit stronger. But I can't show it. Can someone help me please?
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1answer
254 views

boundedness of an operator

Define $T: L^2(\mathbb{R})\to L^2(\mathbb{R})$ by $(Tf)(x)=\int_{\mathbb{R}}\frac{f(y)}{1+|x|+|y|}dy$. Is this operator bounded? If it is, then is it also compact? I got stuck in simply applying ...
6
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2answers
179 views

Stone-Čech via $C_b(X)\cong C(\beta X)$

I am having some trouble constructing the Stone-Čech compactification of a locally compact Hausdorff space $X$ using theory of $C^*$-algebras. I did some search but could not find a good answer on ...
3
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2answers
47 views

$ \left( ||T^n||\right)^\frac1n \leq ||T|| $ as $n\to\infty$?

Given an operator $T\in B(X,X)$, I'm trying to prove that $$\lim_{n\to\infty} \left( ||T^n||\right)^\frac1n \leq ||T|| $$ I can show that $||T^n||\leq||T||^n$, but only for finite $n$. How can I be ...
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0answers
30 views

Density of finite element functions in $W^{1,p}(\Omega)$

I would like to know if the following statement is true: For each $u \in W^{1,p}(\Omega)$ and $\varepsilon > 0$ there exists a piecewise affine function $u_{\varepsilon}$ and a triangulation of ...
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0answers
135 views

Unbounded operator on $C[0,1]$

It is well known that the differential operator is an unbounded operator on the space of all continuously differentiable function on $[0,1]$. However,I found difficulties in finding an unbounded ...
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2answers
803 views

Exercise: Application of Hahn-Banach Theorem

I'm working on this exercise (not homework) and I would gladly welcome some hints for how to solve it! Exercise: Let $\{x_1,\dots,x_n\}$ be a set of linearly independent elements of a normed vector ...
0
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1answer
365 views

Minimizing sequence

This came up in a proof I was reading. Define $$\inf_{z \in K} \|x-z\| = d$$ Let $y_n\in K$ be a minimizing sequence How do we know that such a minimizing sequence exists? Here K is a closed convex ...
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1answer
237 views

How can I calculate the weak derivative of $\frac{1}{\sqrt{x}}$?

Define a function $f(x)$ on $\mathbb{R}$ as: $$f(x) =\begin{cases}0&\text{for }x \le 0\\\\\frac{1}{\sqrt{x}}&\text{for }x > 0\;.\end{cases}$$ Then, how can I calculate the weak ...
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1answer
90 views

Differential Operator on $L_{2}$ problem

I am working on a problem from a textbook and have run into difficulties on this specific question. Any assistance will be appreciated, Consider the partial differential equation, $\frac{\partial ...
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0answers
91 views

what are conormal distributions?

According to the first answer in this post, a conormal distribution $u$ on a manifold $X$ relative to a (closed, embedded) submanifold $Y$ is an element of a Banach (or Hilbert) space $H$ such that ...
3
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1answer
79 views

Toeplitz Operator question

Let $\chi_1$ be the map on the unit circle defined by $\chi_1(e^{it})=e^{it}$. Let $T_{\chi_1}$ be the corresponding Toeplitz operator. Consider the map $T_{\chi_1}^* T_{\chi_1}- T_{\chi_1} ...
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Application of a result on some bounded functionals on a subspace of $C([0,1])$

The following result was proved in a previous post: Bounded functionals on Banach spaces. Let $(X, \|.\|)$ be a Banach space such that $X \subset C([0,1]) $ For every $r\in \mathbb{Q}\cap[0,1], ...
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270 views

Hilbert transform and Hilbert matrix

The Hilbert matrix is \begin{bmatrix} 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \dots \\[4pt] \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & & \ddots \\[4pt] ...
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Extension of differentiation operator to $L_2[0,1]$.

I'm studying for my functional analysis exam. We are required to know the proof of the following, but I cannot figure it out. Consider $L_2[0,1]$ with orthonormal basis $(e_n)_{n=-\infty}^\infty$ ...
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1answer
61 views

Finding the minimizing vector of a $l_{2}$ sequence

I am working on a problem sheet and this question has me stuck. A little guidance will be appreciated. Let $X = l_{2}$. Let $x \in X$ be given by $x = \{\frac{1}{2^{i}} \}^{\infty}_{i=1}$ Let $M ...
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1answer
106 views

Topology and limit points

In my functional analysis homework, I had to prove something like this : Let $D_n \subset D_{n-1} \subset \dots \subset E$ where $(E, \mathfrak T)$ is a Hausdorff topological space and the sequence ...
3
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1answer
116 views

Are the constant functions a closed subspace in the polynomials?

Consider all polynomials $\mathbb R[x]$ and the subspace of polynomials of degree $0$, which we will refer to by the letter $U$. Is this subspace closed with respect to the inner product: $$\langle ...
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1answer
51 views

Operator defined via a sequence of weights

Let the linear operator $T:l^2\rightarrow l^2$ be defined by $y=Tx$ where $x=\{\xi_j\}$, $y=\{\eta_j\}$, and $\eta_j = \alpha_j \xi_j$, where $\{\alpha_j\}$ is a dense sequence in $[0,1]$. Does ...
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1answer
318 views

Every Hilbert-Schmidt is an integral operator?

Let $(X,\mu)$ be a $\sigma$-finite measure space. If $K\in\mathcal{L}^2(X\times X,\mu\times\mu)$ then the map $A_K:\mathcal{L}^2(X,\mu)\to\mathcal{L}^2(X,\mu)$ defined by\begin{equation} ...
3
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1answer
290 views

cyclic vector exists for symmetric operator iff there no repeated eigenvalues

Considering a symmetric operator $A$ acting on a finite dimensional Hilbert space $H$, we say $x\in H$ is a cyclic vector for $A$ if the set of finite linear combinations of $\{A^n x:n=0,1,2,...\}$ is ...
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2answers
247 views

Weak convergence implies $L^p$ convergence on a smooth bounded domain?

Let $U\subset \mathbb{R}^n$ be a bounded domain with smooth boundary. Let $f_k\in L^p(U)$. Does weak convergence of $f_k$ to $f \in L^p$ implies $L^p$-convergence of $f_k$ to $f$? By weak convergence ...
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0answers
94 views

Sampling Theorem Poisson Formula

Theorem If the Fourier transform $\hat{f}(w)$ of a signal function $f(x)$ is zero for all frequencies ouside the interval $-w_c\leq w \leq w_c$, then $f(x)$ can be uniquely determined from its sampled ...