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

Function Singularity in a Sobolev Space

For which $t$ and $p$ does function of the form $$ f(x) = 1/|x|^{\alpha} ,\quad 0<\alpha<1$$ belong to the fractional Sobolev space $H_p^t(\mathbb{R})$, $p>1,t\geq 0$? UPD: actually I am ...
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3answers
50 views

partial differential equation-exercice

let in $\mathbb{R}^2$ the equation $$ \dfrac{\partial^2 u(x,t)}{\partial t^2} - \dfrac{\partial^2 u(x,t)}{\partial x^2} = 0 $$ We put: $ \begin{cases} x=\xi + \eta\\ t=\xi- \eta \end{cases} $ and ...
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0answers
20 views

How to compute the gradient of the following matrix function? [on hold]

$f(X)=\left\|XX^T-I \right\|_F^2$, where $\left\|\cdot \right\|_F^2$ is Frobenius matrix norm
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1answer
36 views

eigenvaue of Sturm Liouville problem

Let the limit probem $$ \begin{cases} (P(x) y')' + q(x) y' + \lambda r(x) y=0\\ \alpha_0 y(0)+ \alpha_1 y'(0)\\ \beta_0 y(l) + \beta_1 y'(l) \end{cases} $$ with $\alpha_0^2 + \alpha_1^2 >0$ and ...
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1answer
31 views

Understanding dual spaces and Riesz's theorem

Is this a proper statement for the Dual space of a Hilbert space? Let $H$ be a Hilbert space. The set of all continuous bounded linear maps, $\mathcal{L}(H,\mathbb{R})$, is called the dual space. I ...
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1answer
20 views

Does the pth James space Jp contain a norm-1 basic sequence domimated by l2? Equivalently, is there a noncompact operator from l2 into Jp?

The $p$th James space, denoted $J_p$, is just the regular James space using the $p$-norm in place of the 2-norm. See here for a complete definition. To use their notation, let $\mathbb{N}_0$ denote ...
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1answer
16 views

Examples in $W^{1,p}(U)\setminus C(\overline{U})$ and $C(\overline{U})\setminus W^{1,p}(U)$

The following is the trace theorem in Partial Differential Equations by Evans: Let $U$ be a domain (open connected subset) of $\mathbb{R}^n$. Suppose $U$ is bounded and $\partial U$ is $C^1$. Then ...
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0answers
13 views

Approach a length by a BV norm

Let $\Omega$ be a smooth bounded open domain in $\mathbb R^d$. Let $g: \overline{\Omega}\to \mathbb R^+$ defined by $g(x)=f(x)$ if $x\in \Omega$ and $g(x)=h(x)$ if $x\in \partial \Omega$, where ...
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0answers
16 views

Lower semi continuity for the norm of the speed

Let $\Omega$ be a smooth bounded open domain in $\mathbb R^d$. Let $\gamma_n: [0,1]\to \overline{\Omega}$ be a sequence of Lipschitz functions which converges uniformly on $[0,1]$ to a Lipshitz ...
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0answers
25 views

Proof that inequality holds

Theorem: Let $u \in D'(\Omega)$ and $K \subset \Omega$, $K$ compact $$\exists \lambda \in \mathbb{N} \text{ and } c \geq 0 \text{ such that } \\ |\langle u, \phi \rangle| \leq c \sum_{|a| \leq ...
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2answers
21 views

$f:(X,d_X)\rightarrow (Y,d_Y)$ isometry then $f(x)=g(x)$ for $x\in\Omega\subset X$ dense, $g: X\rightarrow Y$?

Let $(X,d_X)$ be a metric space and $\Omega\subset X$ be dense. Let $(Y,d_Y)$ be a complete metric space and $f:\Omega\rightarrow X$ such that $d_Y(f(x_1),f(x_2))=d_X(x_1,x_2)$. Why do we get a map ...
2
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1answer
17 views

non-Archimedean Valued field extension of $\mathbb{R}$

Let $K$ be a field with non-Archimedean valuation $|\cdot|$. Suppose that $\mathbb{R}\subset K$. Question 1: Is the restriction of $|\cdot|$ to $\mathbb{R}$ the trivial valuation? I guess that the ...
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0answers
17 views

Prove that $K$ is compact self adjoint operator

The integral operator $$K: L^2 [0,1] \to L^2 [0,1]$$ is defined by $$(Ku)(x)=\int^1 _0 k(x,y) u(y) dy$$ Where $$K(x,y)=min\{x,y\}$$ for $0 \leq x, y \leq 1$. How is the operator continuous on ...
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0answers
21 views

Set of Lipschitz Continuous Functions and its Complement are Dense in C[a,b]

We define $M_{K}$ = {$f: [a,b]\to R $ |$\forall x,y \in [a,b]$ $|f(x)-f(y)|\leq K|x-y|$} and let M = $\cup_{K>0} M_{K}$ I want to show that $M$ and its complement are dense in $C[a,b]$. In my ...
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1answer
45 views

Hilbert space …

i got the following operator. Let H be a Hilbert space, $(\lambda_n)_{n \in \mathbb{N}}$ a bounded sequence in $\mathbb{K}$ and $T^: H \to H, x \to \sum_{n\in\mathbb{N}} \lambda_n \left<x, ...
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1answer
20 views

Two different supremum, are they the same?

Let $X$ be a normed linear space, and let $Y$ be a subspace of $X$. Let $l\in X'$, the dual of $X$. Are these two supremum the same? $$\sup_{\|x+y\|\leq 1,x\in X, y\in Y}|l(x+y)|$$ ...
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1answer
34 views

Show $Tx=\sum^\infty _{n=1} \lambda_n \langle x , u_n \rangle v_n$ defines a bounded linear operator

Let $u_n$ and $v_n$ be two orthonormal basis in a Hilbert space $H$ and let $\lambda_n$ be a bounded sequence of complex numbers. Define $$Tx=\sum^\infty _{n=1} \lambda_n \langle x , u_n \rangle ...
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0answers
12 views

integration with delta function

Is there any way to calculate the following expression: $$\{\frac{\partial}{\partial t}\int|(1-t)p(x)+t\delta_{x_0}(x)-c|dx\}_{\text{at t=0}}.$$ Here, $p$ is a probability density function, ...
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1answer
34 views

Two normal operators are similar if and only if they are unitarily similar

I need to prove that in a $C^*$-Algebra two normal operators are similar if and only if they are unitarily similar. Can anybody help, please? One side is obvious, so our concern is the other side. I ...
4
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1answer
31 views

Weak convergence of bounded operators

so let $X$ be a Banach space then we say that $A_n \in L(X)$ converges weakly to $A \in L(X)$ if for all $y \in L(X)^*: y(A_n) \rightarrow y(A).$ On the other hand, I just read that weak convergence ...
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0answers
24 views

Spectrum of Linear Operator

The spectrum of a linear operator on a finite dimensional space is pure point spectrum, that is, both continuous and residual spectrums are empty. Or we can say that on a finite dimensional space, ...
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0answers
19 views

Weak continuity of the duality mapping

Let $X$ be a Banach space, supposed to be reflexif (but not Hilbert), and let $F$ be the duality mapping, supposed to be univoque and Lipschitz. I'm looking for a sufficient condition under which ...
1
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1answer
37 views

Finite Power of Operator Norm

I know that for any bounded operator A on a normed space, we have $||A^n||$ $\leq$ $||A||^n$. I am wondering when the equal sign would be achieved.
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0answers
31 views

Show $c||x|| \leq ||Ax|| \leq d||x||$ where $c$ and $d$ are positive constants

Let $X$ be a Banach space and $A:X \to X$ a continuous linear map which is bijective. Show for all $x \in X$ $$c||x|| \leq ||Ax|| \leq d||x||$$ where $c$ and $d$ are positive constants. This is the ...
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2answers
27 views

Tridiagonal matrix inner product inequality

I want to show that there is a $c>0$ such that $$ \left<Lx,x\right>\ge c\|x\|^2, $$ for alle $x\in \ell(\mathbb{Z})$ where $$ L= \begin{pmatrix} \ddots & \ddots & & & \\ ...
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0answers
31 views

The space of sequences which are eventually zero in $l^2$ is not a Hilbert space.

Define $V$ to be the space of sequences which are eventually zero, i.e. $$V=\bigcup_{N=1}^{\infty}\{(x_n)_{n=1}^{\infty}\in l^2: x_n=0 \; \text{for}\; n\ge N\}.$$ Is $V$ a Hilbert space with ...
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0answers
11 views

Spectral Family [closed]

I am following E. Kreyszig "Introductory functional analysis with applications". I stuck at this point in Chapter 9, section 9.7. I glad if someone help me understand the idea...Please enlighten ... ...
2
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1answer
29 views

non direct sum decomposition using Fredholm operators

I am reading a paper on PDEs and at some point the author uses an argument that I cannot understand very clear, it seems to be elementary but I do not get it. Let $P$ be a Fredholm operator (i.e. ...
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1answer
14 views

Continuity condition in locally convex F - spaces

It is well known that if $X$ and $Y$ are Banach spaces, then if the linear map $T:X→Y$ satisfies the condition $f∘T ∈ X^{*}$ for every $f∈Y^{*}$, then T is bounded. Is the conclusion still true under ...
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2answers
40 views

Approximation of $\int_0^tF_x(s,X_s)Φ_0dW_s$ where $dX_s=φ_sds+Φ_sdW_s$ and $F_x$ is the Fréchet derivative of some $F:[0,t]×H→\mathbb R$

Let $U$ and $H$ be Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ be equipped with the usual inner product $(\Omega,\mathcal A,\operatorname P)$ ...
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2answers
15 views

$\{g^{-1}(\Omega):\Omega\in\tau_Y\}\in \tau_X$ in topology for continuous functions

Let $g: X\rightarrow Y$ be continuous. Let $(X,\tau_X)$ and $(Y,\tau_Y)$ be topological spaces. Why do we have $\tau=\{g^{-1}(\Omega):\Omega\in\tau_Y\}\subseteq\tau_X$? My attempt: Let ...
0
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1answer
34 views

Find the value of a such that F(a) achieves its minimum value

Find the value of a such that $F(a)$ achieves its minimum value. $$F(a)=\int_{0}^{\pi/2} \left|\sin x - a\cos x \right| dx $$ I'm trying to use following fact to solve the problem but then I need ...
0
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1answer
25 views

Where does this inequality come from: $\frac{|f(x) - f(x-h)|}{h} \le ||f'||_{L^\infty(x, x-1)}$?

I came across this inequality today: $$\frac{|f(x) - f(x-h)|}{h} \le ||f'||_{L^\infty(x, x-1)}$$ I realise if we let $h \to 0$ we obtain the derivative on the left hand side so I can see it has ...
0
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1answer
32 views

weak convergence and weak * convergence criterium

I have to solve the following problem Let $X$ be a Banach space. Prove that $x_n\rightharpoonup x$ in $X$ if and only if $\sup||x_n||<+\infty$ there exists a dense subset $E'$ of ...
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0answers
17 views

How to represent the function of variables?

I have a function as $$E=\int_\Omega -\log\big( p_i(x)\big) dx$$ where $p_i(x)$ is density distribution which estimated by Parzen window method. $p_i(x)=\frac{1}{\Omega_i} ...
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1answer
17 views

Find the point spectrum

Let $x$ and $y$ be nonzero vectors in a Hilbert Space $H$. Let $f(z) = \langle z, x \rangle y$. Find the point spectrum of $f$. I know that $\lambda$ is in the point spectrum if $f(z)=\lambda z$ for ...
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1answer
29 views

How to prove a normal operator is reducible?

I want to prove: given a bounded normal operator $T$ on a Hilbert space $H$. If $H_1$ is a $T$-invariant subspace, then the orthogonal complement $W_1$ of $H_1$ is also $T$-invariant. I can prove the ...
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2answers
54 views

Itō formula as presented in “Stochastic Equations in Infinite Dimensions” by Giuseppe Da Prato

In Stochastic Equations in Infinite Dimensions, Theorem 4.32 (Google Books), the authors present the following version of an Itō formula: Given Hilbert spaces ...
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0answers
11 views

Non-symmetric case, can we get the minimization problem in Lax Milgram therom?

In the Dietrich braess's book 'Finite Elements'. The Lax-Milgram thm is stated under the condition that $a$ is symmetric(There are many book which also state property of symmetric when getting a ...
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0answers
18 views

Aproximation in $W^{1,p}(U)$ with U disconnected.

Consider $U=(-1,0)\cup(0,1)$. Define $$v(x)=\left\{\begin{array}{rc} 0,&\mbox{se}\quad -1<x<0,\\1, &\mbox{se}\quad 0<x<1. \end{array}\right. $$ Clearly $v\in W^{1,p}(U)$ for each ...
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0answers
32 views

Is the solution to the heat equation always $C^k$, no matter what the boundary condition is?

Let $\Omega$ be a bounded Lipschitz domain (can be smooth if necessary). Consider the heat equation $$u_t - \Delta u = 0$$ $$u(0) = u_0$$ $$\text{some Robin boundary condition (B) for $u$}$$ We know ...
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1answer
30 views

Finite Dimensional Hilbert Space

A while ago someone asked this question. I really like what the accepted answer is trying to do. But, I am having trouble figuring out his justification for the first line in the proof: $$\bigcup_{x ...
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0answers
34 views

Lebesgue measurable integration, density

Let $\mathbb{T}$ be the unit circle and $\lambda$ be the Lebesgue measure on $\mathbb{T}$. Let $A_n := e^{2\pi i[1/2^{2n},1/2^{2n+1}]}$, $n\ge 1$. Define a function $f$ on the set of all the Lebesgue ...
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0answers
23 views

Mathematical tools for iterative / composed function analysis

Given the following: a. x0 = known scalar b. x1 = known scalar c. x2 = unknown scalar d. an unknown iteratively applied stochastic nonlinear function $\mathbf{g}$ where ...
2
votes
1answer
23 views

Let $A$ be a Banach algebra. Suppose that the spectrum of $x\in A$ is not connected. Prove that $A$ contains a nontrivial idempotent $z$.

While trying to solve the exercise below, I came up with a wrong conclusion, but I can't see why it's wrong. Also I'm accepting suggestions to get the right solution. This is the problem 17 from ...
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1answer
43 views

Continuity of the operator “inverse of” in the space of linear, bounded and bijective operators

Let $(X,\|\ \|_X),(Y,\|\ \|_Y)$ be two Banach spaces over $K$. Definition: $\qquad\qquad\qquad\quad I(X,Y):=\big\{A\in \mathscr B(X,Y):A\text{ is invertible and }A^{-1}\in\mathscr ...
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0answers
24 views

Bounds on $L^2$ and $L^{\infty}$ norms in terms of $H^1$-seminorms for functions attaining a zero in a domain.

We have the following result in one dimension: If $f\in C^1([a,b])$ attains a zero in $[a,b]$, then $$| f |_2 \leq (b-a) | f' |_2$$ and $$ | f |_{\infty} \leq (b-a)^{1/2} | f' |_2$$ Is there a ...
0
votes
2answers
49 views

Why does this proof work: Closed unit ball in $C_0$ is not compact

I know that this question has been asked to death, and multiple solutions are given, but I still don't understand why the "standard" proof works Following Show that the closed unit ball $B[0,1]$ in ...
1
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1answer
40 views

Are generic smooth functions analytic?

It is well-known that generic continuous functions are differentiable almost nowhere. I was somewhat surprised to learn in my functional analysis course that the same is true of $\alpha$-Holder ...
4
votes
0answers
41 views

Functorial construction of the convolution algebra of measures on a group

Let $G$ be a lcoally compact group and $C_c(G) = \lim_{K \subset G} C(K)$ its space of continuous functions with compact support endowed with the topology of the limit of banach spaces $C(K)$ with $K$ ...