Tagged Questions

31 views

42 views

Space of Distribution wrt to topology of uniformly convergence on bounded sets not Frechet-Space.

I found a state, that the Space of Distribution on (here:) $\mathbb{R}^{n}$, which is equipped with the topology of uniformly convergence on bounded subsets is not a Frechet Space. As far as i can ...
23 views

Various convergences in the space of bounded operators

Could you please help me to find some classical (counter)examples in functional analysis? Let $X$ and $Y$ be some normed spaces over $\mathbb{C}$. By $\mathcal{B}(X,Y)$ we denote the space of bounded ...
59 views

43 views

If $\|T\| < 1$, then $I-T$ is invertible and $\|(I-T)^{-1}\| \leq (1-\|T\|)^{-1}$

This is a hint in my functional analysis book, and I cant uncipher it. They give as additional information that $T \in B(X)$ where $X$ is a normed linear space. I think $X$ should be a Banach Space, ...
39 views

About a spectrum of a C*-algebra

Let $A$ be an unital commutative C*-algebra. Show that the spectrum of $A$ is disconnected iff there is a projection $p \in A$ not trivial.
30 views

Point-wise convergence cannot be normed

Let $X$ be an arbitrary set. Consider the space $\mathbb{C}^X$ of all functions $X\to \mathbb{C}$. For each $x\in X$ we build a seminorm $||\cdot||_x$ such that $||f||_x=|f(x)|$. I would like to prove ...
40 views

I have a small question I have that $\lambda_1 ||v_0||^2_{L^2(0,1)}=||v_0||^2_{H^1_0(0,1)}$ and that $v_n\rightarrow v_0$ on $L^2$ (strongly) From the Poicaré inequality i have that ...
66 views

Proof of a theorem about Baire categories

Problem: prove that the set of $C([0, 1])$ functions whose derivative is defined at every point (and it is either finite or infinite) is of the first Baire category. I have no idea how to approach ...
34 views

Operator compact on $H^1 (0,\pi)$

Consider the operator $K\colon H^1(0,\pi)\to H^1(0,\pi)$ defined by duality (Riesz. Theorem) as $$\langle K\phi,\psi\rangle = \int_{0}^{\pi}{\phi(x)\psi(x)\,dx}$$ for all $\psi \in H^1(0,\pi)$, ...
25 views

I have a function$f$ such that: $f:\Omega\times \mathbb{R}\rightarrow \mathbb{R}$ is continuous , there exist $C>0$ and $\theta>2$ such that $|f(t,u)|\leq C(1+|t|^{\theta-1}~ a.e t\in ... 0answers 36 views Question concerning$\limsup$I have this two hypothesis where$q\geq 1$and where$F(x,t)=\int_0^t f(x,s)ds$, p=2 I dont understand how they find this (3.5) Please help me thank you 0answers 53 views Volterra operator with continuously differentiable Kernel has no Eigenvalue First I'll describe the entire question, as it stated in the exercise: let$K(t,s)\in C([0,1]^2$), continuously differentiable in the first coordinate (meaning$K_t(t,s)\in C([0,1]^2$). And let ... 1answer 78 views Rudin's 'Principle of Mathematical Analysis' Problem 7.12 Suppose$g$and$f_n$($n = 1,2,\ldots$) are defined on$(0,\infty)$, are Riemann-integrable on$[t,T]$whenever$0 < t < T < \infty$,$|f_n| \leq g$,$f_n \rightarrow f$uniformly on every ... 1answer 35 views Writing a linear functional as a linear combination Let$X$be a NLS and$f,f_1,\ldots,f_n$be linear functionals on$X$. Let$\bigcap_{i=1}^n\ker f_i \subseteq \ker f$. Show that$f=\sum_{i=1}^n a_i f_i$for some scalars$a_i$. 2answers 44 views Question about Embedding I have two spaces $$H^1((-\infty,+\infty))=\lbrace u, u\in AC, u'\in L^2\rbrace$$ with the norm$||u||^2=\int_{-\infty}^{+\infty} u'^2+\int_{-\infty}^{+\infty} u^2$and ... 1answer 70 views I dont understand this notation Im having a homework question that goes like this:$X$is a Hilbert space, a complete inner product space, show that$B(X)$is not a Hilbert space. My only question for now is what does$B(X)$... 0answers 12 views Second derivative of Impulsive boundary value problem I have this Impulsive problem : $$\begin{cases} -(p(t)u'(t))'=f(t,u(t))\\ u(0)=u(+\infty)=0\\ \Delta(p(t_j)u'(t_j))=h(t_j)I_j(u(t_j)) \end{cases}$$ and the associated functionnal is given by: ... 1answer 87 views Euler's Refutation of Fermat's Conjecture Fermat postulated that all numbers of the form $$2^{2^n}+1$$ are prime (where n = any integer). Then Euler came along with a rather ingenious proof that this was not, in fact the case. I came across ... 1answer 49 views Weak convergence and strong convergence in$L^1$. Suppose that$\Omega$is a Lebesgue measurable set，$f_n \rightharpoonup f$in$L ^1(\Omega)$and$\|f_n\|_{L^1(\Omega)}\rightharpoonup\|f\|_{L^1(\Omega)}$, then can I say that$f_n → f$strongly in ... 0answers 21 views Functional and operator associated to a problem I have a this functional: associated to the impulsive problem : $$\begin{cases} -(p(t)u'(t))'=f(t,u(t))\\ u(0)=u(+\infty)=0\\ \Delta(p(t_j)u'(t_j))=h(t_j)I_j(u(t_j)) \end{cases}$$ ... 0answers 35 views the principle of uniform boundedness If$\{x_n\} \subset \ell^1$, then$\sum_{j=1}^\infty x_n(j)y(j)\to 0$for every$y\in c_0$iff$\sup_n||x_n||_1< \infty$and$x_n(j)\to 0$for$j\geq 1$. I can proof it by the principle of uniform ... 1answer 40 views Show norms are equiv. on$C^1[a,b]$:$\Vert f\Vert _1=\Vert f \Vert_{\infty}+\Vert f' \Vert_{\infty},\Vert f \Vert_2=|f(a)|+\Vert f' \Vert_{\infty}$Here is what I got as a proof. My question is at the end. Thanks On$ C^1[a,b]$we have the norms $$\Vert f\Vert _1 = \Vert f \Vert_{\infty} + \Vert f' \Vert_{\infty},\quad \Vert f \Vert_2 = |f(a)| + ... 1answer 53 views Relations between normed spaces Is the application$$ Id:( C([0,1]), \|\cdot\|_{\infty})\to ( C([0,1]), \|\cdot\|_{1}) $$open? where Id(f)=f, \|f\|_{\infty}=\sup\|f(x)\| and \|f\|_1=\int |f(x)|dx 1answer 23 views Bases in Hilbert space There's a theorem that states that having Hilbert space H, orthonormal basis \{x_n\}, and a set of linearly independent unit vectors \{y_n\}, such that \sum\limits_{n=1}^{\infty}\|x_n - ... 1answer 42 views When is the closed unit ball B^* in the dual space strictly convex? I'm finding the conditions (on the primal normed space X or on the closed unit ball B of X) to ensure that the closed unit ball B^* in the dual space X^* is strictly convex. Anyone can help ... 1answer 17 views Periodicity and period of a function The question is : Let f(x) be a real valued function defined for all real numbers x such that for for some fixed real number a>0, f(x+a)=\frac{1}{2} + \sqrt{f(x)-(f(x))^2} and \frac12\le ... 2answers 27 views T bounded linear, show A_2T=TA_1 for A_2, A_1 compact Let T:H_1 \to H_2 be a bounded linear map between two infinite dimensional Hilbert spaces and suppose that T is both surjective and injective. Let A_2 \in K(H_2) (where K(H) denotes the set ... 1answer 35 views \|f'(x)\|_{L^p} \le C \|f(x)\|_{L^p}^{1/2} \|f''(x)\|_{L^p}^{1/2} for smooth f with compact support I'm trying to prove the following Let f: \mathbb{R} \to \mathbb{R} be a smooth function supported on [a, b] where -\infty < a < b < \infty. 2 \le p < \infty. Then$$ ... 0answers 16 views functional analysis findining dist(x,Z) in L2(-pi,pi) The question in my hw was Let Z=span (1,sint,cost), x(t)=t. Find dist(x,Z) in$L_2(-\pi,\pi)$From a lemma that we learned it says if Z is closed and$x(t)=t \notin Z$then ... 1answer 35 views prove f is bounded linear functional Let$X=C[-1,1]$and define$f:X\rightarrow R$by$f(x)=\int_{- \ 1}^{\ 0} x(t) dt-\int_{ \ 0}^{\ 1} x(t) dt$. Show that$f$is a bounded linear functional. 0answers 35 views Calderón-Zygmund$\times$Schwartz$=$Calderón-Zygmund I am in a functional analysis class, and we are being asked to show that if$\eta$is a Schwartz function and$K$is a Calderón-Zygmund distribution, then their product is also a Calderón-Zygmund ... 1answer 117 views i need help to prove this problem(functional analysis) show that the annihilator of a set M in an inner product space X is a closed subspace of X. 2answers 68 views Show that$\lambda \in \sigma(A),\lambda$not an eigenvalue, implies that$\lambda \in \sigma(A + K)$where$K$is compact. Let$A : H \rightarrow H$be a bounded linear map where$H$is a Hilbert space with$\dim H = \infty$. Suppose that$\lambda \in \sigma(A)$but$\lambda$is not an eigenvalue. Let$K : H \rightarrow ...
Let $X$ be the space $\ell_\infty$ of all bounded sequences of real scalars. If $Y$ is the set of all $x\in X$ that have bounded partial sums (1) Can I say $Y$ is a linear space (as a subspace of ...