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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9 views

is it true that $\int\limits_{x_{n}}^xf(z)dz\longrightarrow 0$

If $x_{n}\longrightarrow x$ then is it true that $\int\limits_{x_{n}}^xf(z)dz\longrightarrow 0$? We have that $f\in L_{2}(0,\infty)$ and takes complex values. I think that it is, but why? In fact I ...
1
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0answers
13 views

Can a Norm be Induced by two Different Complex Inner Products?

Let $(X,\|\cdot\|)$ be a normed vector space over $\mathbb{C}$. If $\|x\|=\sqrt{\langle x,x\rangle}$ and $\|x\|=\sqrt{\langle x,x\rangle'}$ for all $x\in X$ where $\langle,\rangle$ and ...
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9 views

How to identify $L^2(Q_T)$ and $L^2(0, T; L^2(\Omega))$ (MEASURABILITY OF FUNCTION)

How can we identify $L^2(Q_T)$ and $L^2(0, T; L^2(\Omega))$ For my understanding: $Q_T:=(0, T)\times \Omega$; DEF1: $L^2(Q_T)=\{u: (0, T)\times \Omega \to \mathbb{R}, \mbox{measurable and} ...
2
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1answer
10 views

One Note about One to one and Surjective of linear functional

I read a note that: if $ f \neq 0$ is a linear functional on H, then f is onto (surjective) and it is not one to one (injective) in general. Why this is true? i think it need advance ...
2
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0answers
20 views

Equivalent definitions of the trace of a Hilbert-Schmidt operator

I am currently reading the book Spectral Methods in Automorphic Forms, and Iwaniec defines the trace operator in a different way than I am accustomed to. Throughout, assume that everything converges ...
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0answers
6 views

How to show a Borel Operator Measure dilates to a Spectral Measure?

Does anyone know a simple proof of the following theorem stating that a positive Borel operator measure $P$ on $\mathbb{R}$ can be written as $V^{\star}EV$ for a Borel spectral measure $E$? ...
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0answers
16 views

I need a finite lower bound for this functional, or to prove that one does not exist.

Let $0\leq g < \kappa$, $\gamma>0$ and let $f_1,f_2, S$ be arbitrary functions of r, with $f_1,f_2\geq 0$ I'm looking for a lower bound on the functional $\mathcal{E} = \frac{1}{2} ...
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0answers
4 views

Existence of minimum in $H^{1,2}(\Omega)$

I am considering a functional $$\mu(\Omega) = \min \{ u \in H^{1,2}(\Omega), \frac{\alpha \int_{\partial \Omega} u^2 ds + \int_{\Omega} |\nabla u|^2}{\int_{\Omega} u^2 dx} \}$$ I want to show the ...
5
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0answers
31 views

Composition of projections has a fixed point in a Hilbert space

Let Let H be a Hilbert space with an inner product ⟨⋅,⋅⟩ : H×H→R, and induced norm $∥⋅∥ : H→R_+$ Let $C_1$ and $C_2$ be closed, convex, nonempty, disjoint subsets of $H$ with at least one of ...
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20 views

Spectral Measures: Spectral Subspaces

Given a Hilbert space $\mathcal{H}$ and let the Lebesgue measure be $\lambda$. Consider a normal operator $N:\mathcal{D}\to\mathcal{H}$. Denote its associated Borel spectral measure by: ...
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1answer
24 views

If $z$ is in an inner product space $X$, show that $f(x)=\langle x,z \rangle$ defines a bounded linear functional $f$ on $X$.

If $z$ is any fixed element of an inner product space $X$, show that $f(x)=\langle x,z \rangle$ defines a bounded linear functional $f$ on $X$, of norm $||z||$. If the mapping $X\to X'$ given $z\to f$ ...
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0answers
33 views

Does this function have a unique fixed point?

Consider the following equations in fixed-point form: $$\mathbf{x}=F(\mathbf{x})=[f_1(\mathbf{x}),f_2(\mathbf{x}),\cdots,f_n(\mathbf{x})]^T$$ where the function $f_i$ is defined as ...
1
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1answer
38 views

what does it means $∥T(1_{[a,b]})∥$

For each $f\in L_{2}(0,\infty)$, we set $Tf:(0,+∞)⟶C$ with $Tf(s)=\frac{1}{s}\int\limits_{(0,s)}f(t)dt$. For each $0<a<b$ i want to show that $∥T(1_{[a,b]})∥_{2}\geq\frac{b-a}{\sqrt b}$. But ...
3
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1answer
30 views

Implications of some sort of $l^2$/uniform convergence

Sorry about the title, but I couldn't really figure out how to describe my problem in one sentence... I'm having some problems with real limits: For $f,g : \mathbb{N} \to \mathbb{R}$ let ...
2
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1answer
21 views

$L^{2}$ convergence, bounded function.

Let $X$ be a metric space and $\mathcal{B}(X)$ be a Borel $\sigma$-algebra on $X$ and $\mu$ be a finite measure on $X$. We consider continuous functions (denoted by $\{f_{n}\}$) on $X$. If $f_{n}\to ...
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1answer
37 views

Analytic skills in applied math

I am an unexperienced undergraduate student just took few basic math classes. And I have found analysis classes really interesting, like basic analysis, measure theory and functional analysis, and ...
1
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0answers
11 views

$\|\nabla f\|_p\leq C (\|\nabla \times f \|_p +\|\nabla \cdot f\|_p)$

Let $f\colon\mathbb{R^3}\to \mathbb{R^3}$ have compact support. The identity $$ -\Delta = \nabla\times\nabla \times - \nabla \nabla \cdot, $$ and two integration by parts shows that $$ \|\nabla f\|_2 ...
2
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0answers
28 views

T is not compact and orthonormal sequence [duplicate]

I want to show that if $\,T$ is not compact then there exists an orthonormal sequence $x_{n}$ and $R>0$ such that $ \forall n\in \mathbb{N}\,\,\,\,\|T(x_{n})\|\geq R$. It is obvious by the ...
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0answers
11 views

Given the frame $B=\{(1,1,0),(0,1,1),(1,1,1),(0,0,1),(0,1,-1)\}$, find (if possible) a (i) $(2,1)$ surgery,

Given the frame $B=\{(1,1,0),(0,1,1),(1,1,1),(0,0,1),(0,1,-1)\}$, find (if possible) a (i) $(2,1)$ surgery, and a $(1,2)$ surgery that produce tight frames. I am confused with the concept of surgery ...
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1answer
41 views

Hilbert Subspaces: ONB

This might be a duplicate. If so, then please let me know. Thanks! Given a Hilbert space $\mathcal{H}$. Consider a dense subspace $\overline{Z}=\mathcal{H}$. Then it provides an ONB: ...
0
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1answer
42 views

Spectral Measures: Riemann-Lebesgue

Given a Hilbert space $\mathcal{H}$ and let the Lebesgue measure be $\lambda$. Consider a Borel spectral measure $E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$. Denote its associated ...
1
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0answers
11 views

Does this gradient map have a closed range?

Let $\mathbb{T}^n$ be $n$-dimensional torus. Let $H^1(\mathbb{T}^n)$ be the Sobolev space of functions in $L^2(\mathbb{T}^n)$ whose weak derivative is in $L^2(\mathbb{T}^n)$. Then the gradient map ...
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0answers
9 views

for every frame $B$ with $k$ vector in $\mathbb{R}^2$ such that $B \cup v$ is a tight frame.

Prove the following: for every frame $B$ with $k$ vector in $\mathbb{R}^2$ such that $B \cup v$ is a tight frame. Is the same statement true in $\mathbb{R}^3$? Through the discussion provided in ...
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0answers
7 views

A Fredholm alternative for nonlinear operators?

There is a Fredholm alternative of the form: Let $K$ be a compact linear operator. Then $(I + K)u = f$ has a solution $u$ for every $f$ if and only if $$\text{$(I+K)u=0 \implies u=0$.}$$ Is ...
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0answers
24 views

Understanding integration and substitution

I'm an undergraduate student in EE. I often see that when talking about voltages, curents ... being expressed like functions of some independed variable (time) and when calculating integrals people ...
0
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1answer
19 views

Show that A is unitary

I'm trying to show that $S+i(I-S^2)^{1/2}$, where $S$ is a self adjoint matrix of norm $\leq 1$, is unitary. I have already checked that $I-S^2$ is positive. I am aware that I need to use the ...
1
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1answer
15 views

Reducing Subspaces: Nonexample?

Given a Hilbert space $\mathcal{H}$. Consider an operator $T:\mathcal{D}(T)\to\mathcal{H}$. Suppose there exists a closed subspace $Z\leq\mathcal{H}$: $$TZ\subseteq Z,TZ^\perp\subseteq Z^\perp$$ ...
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0answers
37 views

Additivity of Lebesgue integral w.r.t. sets on non-finite domain

I know that for any Lebesgue integrable function $f:X\to\mathbb{C}$, or $f:X\to\mathbb{R}$, where $X$ is a set of finite measure such that $X=\bigcup_n A_n$, $\forall i\ne j\quad A_i\cap ...
0
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0answers
20 views

Does natrual numbers are isomophic to integers? [on hold]

If there exists a natural numbers algebraic structures , N is the set of natural numbers, which is equipped with the addition operation on it. For another integers algebraic structure, Z is the set of ...
0
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1answer
7 views

Ordered Projections: Range

Given a Hilbert space $\mathcal{H}$. Consider two orthogonal projections $P,Q$. Then: $$P\leq Q\implies\mathcal{R}(P)\subseteq\mathcal{R}(Q)$$ The ordering being induced by: ...
1
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0answers
31 views

The operator Tf(x)=1/x∫f(t)dt on L2 is not compact [on hold]

Let $H=L_{2}(0,+\infty)$. For each $f$ define $T_{f}:(0,+\infty) \longrightarrow \mathbb{C}$ with $T_{f}(x)=\frac{1}{x}\int\limits_{0}^{x}f(t)dt$. I want to show that i) $T_{f}$ is continuous ...
3
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1answer
19 views

Sequence of bounded linear functionals on $C^1[0,1]$ that shows Principle of Uniform Boundedness fails without completion.

Let $X$ be the normed vector space $C^1[0,1]$, of continuously differentiable functions on $[0,1]$ with the sup norm $\displaystyle \|f\|=\max_{t\in[0,1]}|f(t)|$. Find a sequence of bounded linear ...
-2
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1answer
27 views

a question about contractions on Hilbert spaces

Let $\cal{H}$ be a Hilbert space, $T_1,T_2\in\cal{B(H)}$, $\|T_1(h_1)+T_2(h_2)\|^2\leq\|h_1\|^2+\|h_2\|^2$ for all $h_1,h_2\in\cal{H}$. $T_1T^\ast_1+T_2T^\ast_2\leq I$. Then can we verify that 1 ...
1
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1answer
29 views

In a Hilbert Space, if $\langle x,x_n \rangle \to 0$ then $\sup\{\|x_n\|:n=1,2,3,…\}<\infty$

Let $\mathbb{H}$ be a Hilbert space. Let $\{x_n\}$ be a sequence in $\mathbb{H}$ with the property that $\langle x,x_n \rangle\to 0$ as $n\to\infty$ for $x\in\mathbb{H}$. Show that ...
0
votes
1answer
22 views

$X,Y$ are Banach spaces, $T$ is linear, $x_n\to 0$ and $Tx_n\to y$, then $y=0$ and $T$ is continuous.

Here is the question I have: Let $X,Y$ be Banach spaces and $T:X\to Y$ be linear. Suppose that whenever $x_n\to 0$ and $Tx_n\to y$, then $y=0$. Show that $T$ is continuous. So this is what I have: ...
4
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3answers
34 views

A direct proof on if $(X, ||\cdot ||)$ is a normed vector space and $Y\subset X$, with $Y$ having finite dimension, then $Y$ is closed.

I am trying to produce a direct proof on the statement mentioned above. The field I am working in is $\mathbb{R}$. My proof outline goes as following: If $Y$ is finite-dimensional, there exists a ...
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0answers
19 views

Polynomial density in $L^p (\mathbb{R},\mu)$

I wanna check a necessary and sufficient condition for a Radon measure witch have the moments of all orders, to say that polynomials are dense in $L^p (\mathbb{R},\mu)$. Or just a paper or an article ...
3
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1answer
42 views

Normal Operator: Everywhere defined implies bounded?

Given a Hilbert space $\mathcal{H}$. Consider a normal operator $N:\mathcal{H}\to\mathcal{H}$. If its domain is the whole Hilbert space then is it necessarily bounded? The point is that I'm trying ...
1
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2answers
41 views

bounded operator $T$ is not compact then there exists an orthonormal sequence $e_n$ and $d>0$ such that $\|T(e_n)\|>d$ for all $n\in\Bbb{N}$?

I want to prove that if a bounded operator $T$ is not compact then there exists an orthonormal sequence $e_n$ and $d>0$ such that $\|T(e_n)\|>d$ for all $n\in\Bbb{N}$. Could someone helps me?
0
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0answers
13 views

How to deal with $\nabla(\delta\Psi)$ in functional derivatives?

I am trying to compute the functional derivative of the following functional $F[\Psi]=\int{}d^nx\Psi{}e^{(\nabla\Psi)^2}$ what I have tried up till now is the following ...
0
votes
1answer
42 views

Finding the norm of integral operator

I have the following operator: $A: (C^1[0;1];|||\cdot|||)\rightarrow(C^1[0;1];||\cdot||_{\infty})$ $Af(x)=\int_0^x f(t)dt$ where $|||f|||= ||f||_\infty+||Af||_\infty$ What is $||A||?$ I wrote ...
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0answers
10 views

Existence and uniqueness of Volterra integral equations of the first kind with vanished kernel

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To prove the existence and uniqueness of solutions of Volterra integral equation(VIE) of the first kind, we usually differentiate it and convert to the VIE of the ...
1
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0answers
11 views

Giving a bound for |f(x) \star \phi_k(x) -f(x)|

Here is the problem: Let $\phi(x) \in S$, where $S$ is the Schwartz class, such that $\displaystyle\dfrac{1}{\sqrt{2\pi}}\int_{\mathbb{R}} \phi=1$. Also, for some $N\in\mathbb{N}$, ...
0
votes
1answer
18 views

Random variables with all moments. Is this statement true?

Let $X$ be a random variable such thta $X\neq 0$, $P$-a.s. Then $$X\in \bigcap_{p\geq 1} L^p(\Omega) \iff \frac{1}{X} \bigcap_{p\geq 1} L^p(\Omega).$$ In other words, is the space $\bigcap_{p\geq 1} ...
1
vote
1answer
23 views

Determinant of solution of linear equation

Is there a direct way or method to know if the solution to a linear ODE is invertible? I mean, let $A(t)$ be a ($n$ times $n$) matrix and denote by $X(t)$ an unknown Matrix (of the same dimensions) ...
0
votes
1answer
34 views

Can a continuous linear form have a norm of infinity?

We know that a linear form $A \in V^{*}$ is continuous iff $$ \exists C: C\in R, ||Av|| \leq c ||v|| \forall v \in V $$ but we know too that $$ ||A|| = min\{c>0:||Av|| \leq C||v|| \forall v\in ...
-1
votes
0answers
29 views

Strong convergence in Banach spaces [on hold]

Prove: $X,Y$ are Banach Space, $T_n\in L(X,Y)$, $T_n$ is strong convergent if and only if: (1) $\{||T_n||\}$ is bounded; (2) $D$ is a dense subset of $X$,and $\forall x\in D$,$\{T_nx\}$ is ...
1
vote
0answers
28 views

Applying Stone Weierstrass to this isometry of $C^\ast$-algebra

I proved the following theorem but I'd like to confirm the last part of my proof. Statement: Let $A$ be a non-zero commutative $C^\ast$ algebra. Then $\varphi : A \to C_0 (\Omega(A))$ defined by $a ...
1
vote
1answer
24 views

Image of $C^\ast$-algebra is closed?

Let $A$ be a non-zero commutative $C^\ast$ algebra and let $\varphi : A \to B$ be a homomorphism of star algebras. Please could someone help me how to show that $\varphi(A)$ is closed in $B$?
0
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0answers
20 views

Proving a Sobolev-type ineqauality

Given $I=(0,1)$ and $u\in W^{2,p}(0,1)$ for $p>1$. I am trying to prove that for any $\epsilon>0$, the following hold: $$ \|u\|_{L^\infty(I)}+\|u'\|_{L^\infty(I)}\leq ...