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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If a sequence of self-adjoint linear operators is convergent, show that its limit is self-adjoint.

If a sequence of self-adjoint linear operators is convergent, show that its limit is self-adjoint. I'm lost on this problem. I don't know how to even start this. Any solutions or hints would be ...
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27 views

Calculating the local minimum of a function

Regarding this function $f(x,y)=1007x^2-x^{2014}+(e^y-1+2x^2)^2$. I want to find the strict local minimum of $f$. I started calculating $\nabla$: $\nabla (1007x^2-x^{2014}+(e^y-1+2x^2)^2)$ ...
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0answers
10 views

Wave Operators: Adjoint

Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}H_\#\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
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1answer
29 views

On the space $l_2$ we define an operator $T$ by $Tx=(x_1, {x_2\over2}, {x_3\over3}, . . . )$. Show that $T$ is bounded, and find its adjoint.

On the space $l_2$ we define an operator $T$ by $Tx=(x_1, {x_2\over2}, {x_3\over3}, . . . )$. Show that $T$ is bounded I know that $||T||\leq 1$, but I don't know how to show this. Any solutions or ...
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1answer
18 views

Projections: Ordering

Given a unital C*-algebra $1\in\mathcal{A}$. Consider projections: $$P^2=P=P^*\quad Q^2=Q=Q^*$$ Order them by: $$A=A^*:\quad A\geq0:\iff\sigma(A)\geq0$$ Then one has: $$P\leq Q:\quad P=PQ=QP$$ How ...
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3answers
44 views

Definition in Lax “sequence of continuous functions tending to $\delta$”, are distributions needed for understanding?

I'm trying to read Lax's functional analysis. In chapter 11 he makes a definition which I don't like. A sequence of continuous functions ${k_n}$ on a $[-1,1]$ tends to $\delta$ if $\int_{-1}^{1} ...
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0answers
14 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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18 views

question about $L^2([0,T]\times \Omega)$ and iterated integral

Let $X=[0,T]\times \Omega$ where $\Omega$ a bounded domain. Consider the space $L^2(X)$, so $u \in L^2(X)$ if $$\int_{[0,T]\times\Omega}|u|^2 < \infty.$$ Is it true that the integral $$\int_0^T ...
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1answer
31 views

Is the set defined by inequality $\|Tx\|^2\leq\|T^2x\|\|x\|$ a subspace of a Banach space?

Let $X$ be a complex Banach space and $T$ be a bounded linear operator on $X$. Put $Y=\{x\in X:\|Tx\|^2\leq\|T^2x\|\|x\|\}$. Is $Y$ a subspace of $X$? I know is that $Y$ is closed and $aY$ is ...
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0answers
11 views

Wave Operators: Cook

Problem Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}H_\#\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote for shorthand: ...
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3answers
32 views

Show that the operator sequence $ A_n = 1/2(A_{n-1} + A^{-1}_{n-1})$ converges strongly, $A_0 = I+T$, where $T$ is compact and $||T|| \le 1/2$.

I'm studying for an analysis prelim and am stumped on an old exam problem for which there are no solutions given. The full question is as follows: Let $X$ denote a Hilbert space, and $T$ a ...
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0answers
36 views

How could one invert this sum of Stirling numbers?

In this paper, under Stirling Numbers and their Asymptotics, the author takes equation (3.1): ...
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1answer
20 views

Show that an operator of rank $n$ can have at most $n$ nonzero eigenvalues.

Show that an operator of rank $n$ can have at most $n$ nonzero eigenvalues. I'm not sure how to proceed. I think induction will be the best. Any solutions or hints are greatly appreciated.
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Hamiltonian: Compactness

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote the resolvent: $$R(z):=(z-H)^{-1}\in\mathcal{B}(\mathcal{H})$$ Denote compact ...
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0answers
28 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 ...
2
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2answers
25 views

Is a norm closed set(in the topology induced by the norm) weakly closed?

My attempt:Yes because since $T_{\text{Norm}} \supset T_{\text{Weak}} \implies T_{\text{Norm}}^{C} \subset T_{\text{Weak}}^C$ Right? Or have I got something wrong here? This first set inclusion ...
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5answers
847 views

If a unit ball is compact then why a ball of radius 5 has to be compact too?

So if I use the definition of compactness that every open cover has a finite sub-cover, then as the unit ball is compact , there exists a finite subcover. But if I increase the radius of the ball, why ...
2
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1answer
22 views

Example of Non-separable stochastic process.

This question is related to the link: http://www.encyclopediaofmath.org/index.php/Separable_process The link provided a basic definition of separable Stochastic process. I felt all the process under ...
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2answers
22 views

Essential supremum via cumulant

Let $p(t)=\log \mathbb{E}[\exp (tX)]$ for $X$ real valued random variable. Now it holds (assuming that $p$ is smooth and finite on $\mathbb{R}$) that $p'(\infty)=\text{ess}\sup X$. How can I prove ...
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11 views

Are these two statements involving null sets and $L^2$ Bochner functions equivalent?

Suppose I have two functions $f, g \in L^2(0,T;L^2(\Omega))$ where we have some bounded domain $\Omega$. Suppose that $$\text{for almost all $t$,}\quad f(t) \leq g(t) \quad\text{almost everywhere in ...
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2answers
32 views

Find the rang of $\sin (a) + \sin (b)$ [on hold]

If : $a+b=\frac{\pi }{2}$, Find the range of $$\sin (a) + \sin (b)$$
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22 views

Semigroup solution via Lumer-Phillips

Let $a$ be a coercive, bounded bilinear form on $H^1(\Omega)$, where $\Omega$ is some sufficiently "nice" region. I defined an operator $A:H^1(\Omega)\mapsto H^1(\Omega)^*$ by: $$ (Au)[v]=a(u,v)\quad ...
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0answers
27 views

Singular Spectrum: Criterion

Problem Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ And its spectral measure: ...
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0answers
18 views

principle maximum for the heat equation with Dirichlet conditions

Let us consider the Laplacian operator in a domain $\Omega\subset R^n$, with Dirichlet boundary conditions. For all $f\in L^2(\Omega),$ we denote by $S(t)f$ the solution of the equation $$ ...
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0answers
12 views

Spectral Measures: Permutability

Given a Hilbert space $\mathcal{H}$. Consider spectral measures: $$E^{(\prime)}:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Denote their operators by: ...
1
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1answer
73 views

The set of continuity of a pointwise limit of continuous functions

Let $\{x_n(t)\}_{n=1}^{\infty}$ be real a sequence of continuous function from $[0,1]$ to $\mathbb{R}$, and $\{x_n(t)\}_{n=1}^{\infty}$ converges pointwise to $x(t)$ i.e. $\lim_{n \to \infty} x_n(t) ...
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0answers
11 views

Bounded v.s. completely bounded homomorphisms between $L^p$ operator algebras

Take an $L^p$ operator algebra to mean a closed subalgebra $A\subset B(L^p(X,\mu))$ for some ("nice") measure space $(X,\mu)$, $p\in[1,\infty)$. Equip the matrix algebra $M_n(A)$ with the norm ...
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1answer
21 views

Operator equation $Au = f$ for $-\Delta u(x)=f(x)$

We consider the boundary value problem on a bounded, open domain $\Omega \subset \mathbb R^n$ determining $u : \Omega \rightarrow \mathbb R$ such that $$-\Delta u(x)=f(x), \qquad ...
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0answers
11 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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1answer
21 views

What modification is this of the notion of Renyi divergence?

Given two probability distributions $P$ and $Q$ over the same outcome and event space (assume finite if needed) one defines their Renyi divergence as $D_\alpha (P \vert \vert Q) = \frac{1}{\alpha -1} ...
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1answer
17 views

Number of equivalence Relations containing $(1,2)$

Find the number of equivalence Relations on the Set $A=\{1,2,3 \}$ which contains the Element $(1,2)$. My Try: Since $(1,2)$ is to be included, so is $(2,1)$ since the Relation should be Symmetric ...
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1answer
19 views

how to define that a nonlinear operator is bounded and continuous

We always see the definition of bounded and continuous linear operator. I am wondering how to define that a nonlinear operator is bounded and continuous. Is there any book providing this definition?
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1answer
39 views

Is $C^{\infty}$ dense in $W^{k,p}$?

The $C_c^{\infty}$ are certainly not sense in an arbitrary $W^{k,p}$ space. Despite, I started wondering whether at least $C^{\infty}$ is dense? Now, this can certainly not be true for general ...
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1answer
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How can I prove that $f$ and $g$ are measurable functions [on hold]

Let we have the following functions : $f(x)=(\sin x)^4$ and $g(x)=(\cos x)^4$ How can I prove that $f$ and $g$ are measurable functions
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24 views

Searching for a condition on the derivative $f_u$

Please wht can be the condition on $f_u$ such that we obtain the following equality: $$\int_0^1 \int_0^1 G(t,s)f_u(s,0) v(s) w(t) \ ds\ dt=\int_0^1 \int_0^1 G(t,s)f_u(s,0) w(s) v(t) \ ds\ dt$$ ...
2
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2answers
56 views

Operator on $L^2 (0,1)$ defined by convolution with $|x-y|^{-\alpha}$

Define $A: L^2 (0,1) \to L^2(0,1)$ $$Af(x) = \int_0^1 f(y) \frac{1}{|x-y|^\alpha} dy \quad , \quad \alpha \in (0,1)$$ For what values of $\alpha$ is it well defined? Bounded? Compact? I tried doing ...
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1answer
10 views

Sesquilinear Forms: Polarization

This thread is only Q&A.* Given a Hilbert space $\mathcal{H}$. Consider the transforms: $$q[\varphi]:=s(\varphi,\varphi)\quad s(\varphi,\psi):=\frac{1}{4}\sum_{\alpha=0\ldots3}i^\alpha ...
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0answers
29 views

Sesquilinear Forms: Cauchy-Schwarz

This thread is related: Parallelogram Given a Hilbert space $\mathcal{H}$. Consider a quadratic form: $$q:\mathcal{H}\to\mathbb{C}:\quad q[\lambda\varphi]=|\lambda|^2q[\varphi]$$ Suppose it ...
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1answer
7 views

Spectral Measures: Scale Spaces (V)

This thread is only Q&A. Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}N\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
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1answer
16 views

Spectral Measures: Scale Spaces (IV)

This thread is only Q&A. Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}N\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
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2answers
27 views

Finding adjoint of an operator from $\mathbb{C}^n$ to $H$

Suppose we have vectors $h_1,\ldots,h_n \in H$, where $H$ is a Hilbert space. Define $B : \mathbb{C}^n \to H$ by $$B(z_1,\ldots,z_n)=\sum_{j=1}^n z_j h_j.$$ Calculate $B^* : H \to \mathbb{C}^n$. ...
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1answer
20 views

Spectral Measures: Scale Spaces (III)

This thread is only Q&A. Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}N\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
3
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1answer
34 views

Existence of certain idempotents

Suppose $T$ is an idempotent (that is $T^2=T$) of infinite rank and co-rank on a separable Hilbert space. Can we find an idempotent $S$ such that $\overline{TS(H)}=(Id-S)(H)$?
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1answer
30 views

Spectral Measures: Scale Spaces (I)

This thread is only Q&A. Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}N\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
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0answers
17 views

Regularity of compactly supported solutions to the divergence equation: $\nabla\cdot \mathbf{v}=g$.

I have two questions, one specific and one general, on this result for compactly supported solutions to the divergence equation in star-like domains. The result which can be found in Galdi's "An ...
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3answers
62 views

History of convolution

Let $f, g\in L^{1}(\mathbb R),$ we may define the convolution of $f$ and $g$ as follows: $f\ast g(x)= \int_{\mathbb R} f(x-y)g(y) dy, (x\in \mathbb R).$ It is well known that it can be defined on ...
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0answers
30 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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1answer
19 views

How is completeness used in the Eberlein–Šmulian theorem?

So for weakly closed subsets of Banach spaces compactness and sequential compactness coincide, but upon studying the proof I can't put my finger on what exactly would go wrong if we dropped the ...
1
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1answer
23 views

Does smoothness imply boundedness? Evans PDE chapter 2 Problem 18

In problem 18, enter link description here 1) I am having difficulty in extracting information in deciding the bounded for $g$ and $h$. In particular, to conclude $g$, $Dg$, $h$ $Dh$ are bounded by ...
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2answers
46 views

Is 'a' differentiable in f when f is a product of a differentiable and non-differentiable function?

Recently, I was studying differentiable and non-differentiable functions and I wondered whether this "conjecture" of mine is true: 1) "If $f(x)$ is a function that is the product of $g(x)$ and $h(x)$ ...