Tagged Questions

31 views

Convergence of norms

I have this space $H_{0,p}^1=\lbrace u\in AC([0,+\infty),\mathbb{R}),u(0)=u(+\infty)=0, \sqrt{p} u'\in L^2(0,+\infty)\rbrace$ endowed with the norm $||u||^2=\int_0^{+\infty} p(t) u'^2(t) dt$ ...
45 views

How to find an example

I want to find a function $f\in C^1([0,+\infty)\times\mathbb{R},\mathbb{R})$ such that $f(t,0)=0$ $f(t,u)\leq \alpha u+\beta$, $\alpha<\lambda_1,\beta\geq 0$ $f(t,u)\geq C_1 |u|^{\sigma}$ where ...
60 views

Problem in functional analysis.

I heard of this problem that caught my attention and I am curious now thus I would appreciate if I could have a hint or a solution. Let $(x_n)$ a sequence in a normed space $X$ such that ...
24 views

Biholomorphic, Hypersurface

I'm learning the Hypersurface. And my teacher has a question: Find an example such that two Hypersurfaces are biholomorphic. I think that $$A=\{(x,y)\in \Bbb C,\ \rho(x,y)= x^2+y^2-1=0\}$$ and ...
74 views

Solve the equation: $x(t)-3\int_0^1(s+t)x(s)ds=y(t)$

Given $y\in L^2[0,1]$, Solve the equation: $$x(t)-3\int_0^1(s+t)x(s)ds=y(t)$$ I have noticed that the equation is $(I-K)(x(t))=y(t)$, where $K(f(t))=\int_0^13(s+t)f(s)ds$ is a compact integral ...
23 views

46 views

48 views

Weak solution Boundary Value problem

I have to prove that the following problem $$(P) \begin{cases} -u''-u=1\,\,\,\,\,\,\,\,\text{if}\,\,\, x\in(0,\pi)\\ u(0)=u(\pi)=0 \end{cases}$$ doesn't admit weak solutions. I'm proceeding by ...
39 views

37 views

61 views

How to make a cos function into a sin function

I need to convert this equation into a sin function: f(x) = 12 cos(2x + 1) â 3 I know cos(x)= sin (pi/2 -x) but other than that I dont know how to solve this problem
21 views

Show that $\{w^{1/2}\phi_n\}$ is an orthonormal set in $L^2(D)$ if $\{\phi_n\}$ is an orthonormal set in $L^2_w(D)$

As mentioned in the title, my problem is: Show that $\{w^{1/2}\phi_n\}$ is an orthonormal set in $L^2(D)$ if $\{\phi_n\}$ is an orthonormal set in $L^2_w(D).$ So I know that: ...
28 views

$T$ has a finite rank $\iff$ $\exists N \in \mathbb{N}$ such that $\lambda_n=0$, $\forall n \geq N$

The question goes as follows: $T$ has a finite rank $\iff$ $\exists N \in \mathbb{N}$ such that $\lambda_n=0$, $\forall n \geq N$. Given is the data: $X$ is a Hilbert space with an orthonormal ...
31 views

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
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 ...
81 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 ...
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 $a_i$.
45 views

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 ...
71 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 $B(X)$ ...
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: ...
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 ...
50 views

Weak convergence and strong convergence in $L^1$.

Suppose that $\Omega$ is a Lebesgue measurable setďź$f_n \rightharpoonup f$ in $L ^1(\Omega)$ and $\|f_n\|_{L^1(\Omega)}\rightharpoonup\|f\|_{L^1(\Omega)}$, then can I say that $f_n â f$ strongly in ...
21 views

Functional and operator associated to a problem

I have a this functional: associated to the impulsive problem : $$\begin{cases} -(p(t)u'(t))'=f(t,u(t))\\ u(0)=u(+\infty)=0\\ \Delta(p(t_j)u'(t_j))=h(t_j)I_j(u(t_j)) \end{cases}$$ ...
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 ...
40 views

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}) $$open? where Id(f)=f, \|f\|_{\infty}=\sup\|f(x)\| and \|f\|_1=\int |f(x)|dx 1answer 23 views Bases in Hilbert space There's a theorem that states that having Hilbert space H, orthonormal basis \{x_n\}, and a set of linearly independent unit vectors \{y_n\}, such that \sum\limits_{n=1}^{\infty}\|x_n - ... 1answer 43 views When is the closed unit ball B^* in the dual space strictly convex? I'm finding the conditions (on the primal normed space X or on the closed unit ball B of X) to ensure that the closed unit ball B^* in the dual space X^* is strictly convex. Anyone can help ... 1answer 17 views Periodicity and period of a function The question is : Let f(x) be a real valued function defined for all real numbers x such that for for some fixed real number a>0, f(x+a)=\frac{1}{2} + \sqrt{f(x)-(f(x))^2} and \frac12\le ... 2answers 27 views T bounded linear, show A_2T=TA_1 for A_2, A_1 compact Let T:H_1 \to H_2 be a bounded linear map between two infinite dimensional Hilbert spaces and suppose that T is both surjective and injective. Let A_2 \in K(H_2) (where K(H) denotes the set ... 1answer 35 views \|f'(x)\|_{L^p} \le C \|f(x)\|_{L^p}^{1/2} \|f''(x)\|_{L^p}^{1/2} for smooth f with compact support I'm trying to prove the following Let f: \mathbb{R} \to \mathbb{R} be a smooth function supported on [a, b] where -\infty < a < b < \infty. 2 \le p < \infty. Then$$ ...
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 ...
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.