2
votes
2answers
66 views

Proof of equivalent characterizations of compact operators

As an exercise I tried to prove the following theorem: If $X,Y$ are Banach spaces and $u \in B(X,Y)$ is a bounded linear operator then the following are equivalent: (1) $u$ is compact ...
3
votes
0answers
36 views

Strongly convergence in L^2

Let the sequence $(f_n(x,u))$ such as, for all $n,$ $f_n$ is Caratheodory, and $|f_n|\leq g$ where $g \in L^1(\Omega)$ Let $u_n \in H^1_0$ such as it is strongly convergent to $u$ in $L^2$ and a.e ...
0
votes
0answers
20 views

use Fatou theorem to prouve an convergence

let $u_n$ an sequence uniformaly bounded in $H^1_0(\Omega)$, then, $u_n$ converge weakly to $u$ in $H^1_0$, and strongly in $L^2(\Omega)$ and a.e $x \in \Omega$. Let $g(x,u)$ an Carathedory function ...
0
votes
0answers
56 views

convergence sequence in L^1

Let an sequence $u_n$ such as $u_n$ converge to $u$ in $H^1_0(\Omega)$ weak, and $u_n$ converge to $u$ in $L^2(\Omega)$ strong and a.e $x \in \Omega$. Let $g_n(x,u_n)$ an Caratheodory function such ...
0
votes
0answers
31 views

Application of Schauder fixed point theorem

Let $\Omega$ an open bounded in $\mathbb{R}^n$, and let $A(x,u)$ an Caratheodory function, bounded and coercive, i.e., $$|A(x,u)|\leq \beta$$ $$A(x,u) \geq \alpha I,\quad \alpha > 0$$ and let ...
0
votes
0answers
28 views

Find variational formulation

We have the problem in $\Omega=\mathbb{R}^2_+$ $$ \begin{cases} & -a_{11} \dfrac{\partial^2 u}{\partial x^2} - (a_{12} + a_{21}) \dfrac{\partial^2 u}{\partial x \partial y} - a_{22} ...
3
votes
0answers
53 views

estimations of solutions

Let $\Omega=\mathbb{R}^2_+=\{(x,y) \in \mathbb{R}^2; y > 0\}$, $f \in L^2(\Omega)$, $\lambda \in \mathbb{R}^*_+$, $A=(a_{ij})_{1\leq i,j \leq 2}$, $a_{ij} \in \mathbb{R}$ and there exist $\alpha ...
2
votes
2answers
57 views

Density in $H^1_0$

Let $\Omega = \mathbb{R}^2_+=\{(x,y)\in \mathbb{R}^2; y>0\}$ and let $v \in H^1_0(\Omega)$. For $h \neq 0$ we define $D_h v = \dfrac{v(x+h,y)-v(x,y)}{h}$ such as $\forall \varphi \in ...
0
votes
0answers
82 views

Convolution of piecewise function

I would like to compute the convolution of piece wise function Following is the piecewise function $$ C_a(t) = \begin{cases}0& t\leq t_d\\ ...
0
votes
2answers
84 views

bounded linear functional on $\ell^{1}$, and its relation to $\ell^{\infty}$

Prove that a bounded linear functional $F$ on $\ell^1$ has representation $F(x)=\sum_{n=1}^{\infty}(c_{n}x_{n})$ where $c_{n} \in \ell^{\infty}$, and that $\|F\|_{*} = \|c_{n}\|_{\infty}$.
2
votes
1answer
54 views

Hilbert space, orthonormal system, compact set of vecors

Could you help me solve this problem? Let $e_1, e_2, ...$ be an orthonormal system in a Hilbert space, $\delta_1, \delta_2 ... \in (0, + \infty)$. Prove that the set of all vectors $\sum _{n=1} ...
0
votes
2answers
92 views

$\|.\|_2$ closure of a set which is dense in $L^2[0,2\pi].$

The following is an exercise of Conway's Functional analysis, chapter 1, section 5. Let $L=\{f\in C[0,2\pi]|f(0)=f(2\pi)\}$ and show that $L$ is dense in $L^2[0,2\pi]$.
2
votes
0answers
113 views

Functional analysis problem involving maximal ideal space

For the past week, I've been trying to solve, as a practice homework exercise, Problem 6 of Chapter 11 in Rudin's Functional Analysis, but have not gotten very far it seems. The problem is as follows: ...
1
vote
1answer
73 views

Convolution of distributions is not associative

I need some help with this exercise: It proposes to show that convolution of distributions is not associative: If $T=T_1$ (distribution given by f=1), $S=\delta'$, and $R=T_H$ (we denote as $H$ the ...
0
votes
0answers
29 views

Weak convergence sequence in $\ell^p$ space [duplicate]

Let $e_n = (0,0,\dots,0,1,0,\dots)$ be a sequence, which has $1$ on $n$-th place and zeros elsewhere. Show that $e_n$ weak converges to $0$ in arbitrary space $\ell^p$ for $p \gt 1$, while in $\ell^1$ ...
3
votes
2answers
80 views

exercise on the closed subspaces of an Hilbert spaces

I have a question regarding exercise 3.1.13 of "Analysis Now" by Pedersen volume 118 of the Springer GTM. The exercise aim to show that any closed subspace $X$ of $L^2([0,1])\cap L^{\infty}([0,1])]$ ...
3
votes
1answer
54 views

Turning an isometric embedding into a homeomorphism

While in studying functional analysis, there is a part of a homework problem from Rudin's Functional Analysis that asks to show that the isometric embedding $\phi: X \rightarrow X^{**}$ is a ...
-1
votes
0answers
176 views

Positive elements in a C*-algebra [closed]

Prove that if $a$ is an element in a $C^*$-algebra $A$, then $a$ is positive if and only if $f(a) \geq 0$ for every state $f$ on $A$.
4
votes
1answer
200 views

Prove or disprove existence of a sequence converging weakly to $0$ in an infinite dim Hilbert space

This is a problem on an old analysis qual, the prompt is: "Prove or give a counter example: if $H$ is an infinite dimensional Hilbert space and $0$ is the zero vector in $H$, then there exists a ...
4
votes
2answers
271 views

Show $f_n \rightarrow f$ in $L^2(\mathbb R)$ if $f_n(x) = f(x + x_n)$ where $x_n \rightarrow 0$

This is an old qualifying exam question. My attempt was to say $||f_n - f||_{L^2(R)}^2 = \int_R (f(x+x_n) - f(x))^2 = ||f(x+x_n)|| + ||f(x)|| - 2 \int_R f(x+x_n)f(x) dx$ Now I want to say that ...
3
votes
1answer
129 views

Uniqueness of the involution on a $C^*$-algebra

indication please Let $A$ be a C*-algebra. Suppose that there exists on $A$ another involution $x\rightarrow x^{\#}$ such that $||xx^{\#}||=||x||^2$, for all $x\in A$. Prove that $x^{\ast}=x^{\#}$, ...
1
vote
1answer
97 views

exercise: limit orthonormal sequence, “Banach Space Theory”

I have an exercise from "Banach Space Theory": Suppose $\{x^k\}_{k=1}^\infty$ is an orthonormal sequence in $l_2$, where $x^k:=(x_i^k)$. Show that $\lim_{k\rightarrow \infty} x_i^k =0 \ \forall_{i\in ...
6
votes
1answer
183 views

Maximal ideals in the algebra of continuously differentiable functions on [0,1]

This is an exercise in Rudin's Functional Analysis, in the chapter on commutative Banach algebras. My (uneducated) guess was that every homomorphism on $C^{1}[0,1]$ is an evaluation at some point of ...
1
vote
2answers
179 views

Weak convergence-exercice

Let $\Omega$ be an open set in $\mathbb{R}^n$ and let $(u_n)$ be a bounded sequence in $H^1_0(\Omega).$ Who's the theorem say that we can extract a subsequence denoted $u_{n}$ as $u_n$ weakly ...
2
votes
2answers
247 views

$C^\infty(R^n)$ is a Banach Space when equipped with topology of uniform convergence

Prove $C^\infty(\Bbb R^n)$ is a Banach Space when equipped with topology of uniform convergence. $C^\infty(\Bbb R^n)$ is space of all continuous functions that converge to $0$ at $\infty$. And, the ...
11
votes
3answers
804 views

problem books in functional analysis

There are many excellent problem books in real analysis.I'm looking for a problem book in functional analysis or a book which contains a lot of problems in functional analysis (Easy and hard problems) ...
9
votes
0answers
333 views

Compact set of probability measures

I think I can solve the following exercise if X is assumed to be separable, otherwise I can't. Let $X$ be a (Hausdorff) locally compact space, $\pi\colon X \to Y$ a continuous map into a ...
3
votes
2answers
415 views

Hellinger-Toeplitz theorem use principle of uniform boundedness

Suppose $T$ is an everywhere defined linear map from a Hilbert space $\mathcal{H}$ to itself. Suppose $T$ is also symmetric so that $\langle Tx,y\rangle=\langle x,Ty\rangle$ for all ...