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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19 views

Spectral Measures: Riemann-Lebesgue

Given a Hilbert space $\mathcal{H}$. Consider a Borel spectral measure $E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$. Denote its associated measures by: $$\mu_{x,z}(A):=\langle ...
1
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1answer
30 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 ...
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1answer
23 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 ...
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0answers
18 views

T is not compact and orthonormal sequence

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
10 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 ...
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0answers
7 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 ...
0
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0answers
30 views

Hilbert Subspaces: ONB

This might be a duplicate. If so, then please let me know - I will close this thread then. Thanks! Given a Hilbert space $\mathcal{H}$. Consider a dense subspace $\overline{Z}=\mathcal{H}$. Then it ...
1
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0answers
6 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 ...
1
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0answers
15 views

Measurability of the composition of sections and probability measures

Let $X,Y,Z$ be Polish spaces endowed with their respective Borel sigma algebras, and $\Delta(Y),\Delta(Z)$ be sets of Borel probability measures on $Y$ and $Z$, endowed with the weak*-topologies (and ...
2
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0answers
62 views

Translation invariance and finite dimension imply smoothness

Let $X$ be linear subspace of $C(\mathbb R)$, the set of continuous functions on $\mathbb R$, which is closed under translations, i.e., if $f\in X$ and $h\in\mathbb R$, then $\tau_h f\in X$, where ...
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0answers
6 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 ...
6
votes
3answers
95 views

For which $s\in\mathbb R$, is $H^s(\mathbb T)$ a Banach algebra?

According to Theorem 4.39 in Adams & Fournier Sobolev Spaces: If $mp>n$, $m\in\mathbb N$, then $W^{m,p}(\Omega)$ is a Banach algebra, provided that $\Omega\subset\mathbb R^n$ satisfies the ...
4
votes
3answers
148 views

The equation $\,\,\Delta u+\cos u=0\,\,$ possesses a weak solution in $\,W^{1,2}_0(D)$

Let $D$ be the open bounded smooth subset in $\mathbb{R}^{n}$. Prove that there is a weak solution in $W^{1,2}_0$$(D)$ to following equation $$\Delta u+\cos u=0.$$ Help me some hints to start. ...
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0answers
26 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 ...
4
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3answers
33 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 ...
0
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0answers
6 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
22 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 ...
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0answers
17 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 ...
2
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1answer
86 views

If $u_n \rightharpoonup u\,$ in $\,L^2(\Omega)\,$ and $\,u_n^2 \rightharpoonup v$ in $L^1(\Omega),\,$ then is $v=u^2$?

If $u_n \rightharpoonup u$ in $L^2(\Omega)$ and $u_n^2 \rightharpoonup v$ in $L^1(\Omega)$, then is $v=u^2$? We assume that the domain $\Omega$ is bounded. If not is there any way to ensure this?
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1answer
17 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 ...
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0answers
12 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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2answers
41 views
+50

A question about convex open set in a topological vector space.

Supose $E$ is a topological vector space(may not be Hausdorff). $U\subset E$ is an open set such that $U+U=2U$. How to show $U$ is convex? I can see if $E$ is $T_1$,then $E$ should be Hausdorff. ...
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: ...
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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 ...
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0answers
18 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
votes
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 ...
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1answer
21 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: ...
0
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1answer
26 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 ...
7
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1answer
85 views

The spectrum of a product of operators

Suppose $A,B\in\mathcal{B}(\mathcal{H})$, where $\mathcal{H}$ is an infinite dimensional Hilbert space. In general, we know that there is no relationship between $\sigma(AB)$ and $\sigma(A)$ and ...
2
votes
1answer
40 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 ...
2
votes
1answer
23 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 $B$?
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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 ...
14
votes
2answers
225 views

Finite dimensional subspace of $C([0,1])$

Let $S$ be a subspace of $C([0,1])$, i.e. the continuous real functions on $[0,1]$. Assume that there exists $c>0$ such that $\|f\|_\infty\leq c \|f\|_2$ for all $f\in S$. Then $S$ must be ...
1
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0answers
31 views

On calculating spectral projections

Consider following operator from this paper; Let $h$ be any function in $L^1$ relative to the measure $g(w)dw$ and $K\in\mathbb{C}$ Consider the linear operator $B$ on $L^1$ defined by $$(Bh)(x) = ...
0
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2answers
35 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
votes
1answer
40 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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1answer
33 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 ...
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1answer
44 views

boundary conditions for operator

if you have a Schrödinger operator on a sphere ( $\mathbb{S}^2$) $-\Delta_{\theta,\phi} \psi(\theta,\phi) + V(\theta) \psi(\theta,\phi) = E\psi(\theta,\phi),$ where the potential does not depend on ...
1
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1answer
65 views

Banach fixed point theorem (application)

Let $X$ and $Y$ be Banach Spaces, and $T: X \to Y$, where T is continuous, linear and bijective, and let $S: X \to Y$ (where $S$ is continuous and linear) with $|S|\cdot|T^{-1}|< 1$. Show that ...
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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 ...
1
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1answer
21 views

spectral projection of an element in a C*-algebra

I'm studying Takesaki's Operator theory and I preferred "spectral projection "in the page 43 of this book while he didn't speak about it before. I searched it, but I could not find it. Please explain ...
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0answers
9 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 ...
0
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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} ...
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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 $N\in\mathbb{N}$, ...
3
votes
2answers
53 views

Selfadjointness of the Dirac operator on the infinite-dimensional Hilbert space

I am a physicist, so my background in functional analysis is limited only to basics. However, I would like to prove that the free Dirac operator is selfadjoint (or Hermitian, or neither). The free ...
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) ...
1
vote
1answer
18 views

Natural *-isomorphism

Show that if $X,Y$ are locally compact spaces, then there is a natural *-isomorphism from $C_0(X,C_0(Y))$ onto $C_0(X\times Y)$. My attempt: I define $\phi:C_0(X,C_0(Y))\to C_0(X\times Y)$ such that ...
0
votes
1answer
15 views

Is $C(\Omega)$ a C*-algebra if $\Omega$ is not locally compact, nor compact?

We always say if $\Omega$ is compact or locally compact, then C(\Omega) is a C*-algebra. Now is $C(\Omega)$ a C*-algebra if $\Omega$ is not compact nor locally compact? If not, I want to know which ...
4
votes
1answer
152 views

Completeness of the space of sets with distance defined by the measure of symmetric difference

Let $m$ be the measure defined on the set semiring $\mathfrak{S}_m$ and $m'$ its extension to the minimal ring $\mathfrak{R}(\mathfrak{S}_m)$. I read that $m'(A\triangle B)$ can be used as a distance ...
1
vote
1answer
59 views

Hilbert Space: Weak Convergence implies Strong Convergence

This probably might be a duplicate - let me know if so. I read the following in Graf's notes on quantum mechanics - can you give me a hint for the proof. In Hilbert spaces weak convergence in a way ...