0
votes
0answers
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 $||u||^2=\int_0^{+\infty} p(t) u'^2(t) dt$ ...
0
votes
0answers
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 ...
2
votes
0answers
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 ...
0
votes
0answers
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 ...
3
votes
1answer
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 ...
0
votes
0answers
23 views

Question about convergence

If i have that $$\int_0^{+\infty} a(t)|u_n(t)-u(t)|^2 dt \rightarrow 0 $$ how we can deduce that $$\int_0^{+\infty} a(t)|~|u_n(t)|-|u(t)|~|^2 dt \rightarrow 0 $$ where $a>0, a\in ...
3
votes
1answer
39 views

Show that $\lim_{s \to \infty}F_s(t) = F(t)$ uniformly for $t \in (0,+\infty)$

Given the following functions: $$ F(t)= \int_0^\infty e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t>0$$ $$ F_s(t)= \int_0^s e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t \geq 0, s>0$$ Show that $\lim_{s \to ...
1
vote
1answer
46 views

Need help considering series like these: $\sum_{n=1}^\infty\langle x,e_n\rangle e_n$

I'm working in a Hilbert space $H$ with ONB $(e_n)$ and I have $\alpha=(\alpha_n)\in\ell^\infty$. I have an operator that looks like this: $$T_\alpha x=\sum_{n=1}^{\infty}\alpha_n\langle ...
0
votes
2answers
92 views

Finding the spectrum of this operator

Let $X$ be a Hilbert space and let $\psi_1,\psi_2$ be linearly independent vectors and let $\varphi_1,\varphi_2$ be linearly independent vectors in $X$. Define the operator $T$ in $B(X)$ ...
0
votes
1answer
26 views

Extend the Stone-Weierstrass theorem to high dimension?

I am thinking of if there is high dimensional extension to the well known Stone-Weirstrass theorem. Wikipedia says it is possible to extend the 1D theorem to 2D, i.e. If  f  is a continuous ...
0
votes
1answer
20 views

Showing $\int_{-1}^1\int_{-1}^1(u_x^2+2u_y^2+u^2-x^2y^2u)\, dx\, dy\geq c$.

Prove that for some $c\in\mathbb{R}$: $$G(u) =\int_{-1}^1\int_{-1}^1(u_x^2+2u_y^2+u^2-x^2y^2u)\, dx\, dy\geq c$$ for every $u \in H_0^1$. I know that $$G(u) ...
0
votes
2answers
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 ...
1
vote
1answer
39 views

Application Closed Graph Theorem to Cauchy problem

Consider $E:=C^0([a,b])\times\mathbb{R}^n$ and $F:=C^n([a,b])$ equipped with the product norms. Consider $$ u^{(n)}+\sum_{i=0}^{n-1}a_i(t)u^{(i)}=f $$ with $$u(t_0)=w_1,\dots,u^{(n-1)}(t_0)=w_n \\ ...
1
vote
1answer
31 views

Non-existence of a continous-norm on a sequence space.

For $U\cong \prod_{n\in \mathbb{N}} \mathbb{R}$ equipped with the product topology, i have already shown, that $U$ is a Frechet-Space w.r.t. the frechet-metric. How to prove that there exists no ...
0
votes
0answers
25 views

calculate the sup of the max of 3 functions

Let a function be the variable, how the calculate the following expression? $$\inf_{c(t) \in C[-1,0]} \max \{ \max_{-1 \leq t \leq 0} |c(t)| , \max_{0 \leq t \leq 1} | \int_{0}^{t} c(v-1) +1 dv +c ...
0
votes
0answers
37 views

Green function of Sturm liouville problem

How to find the Green function of the following problem: $$\begin{cases}-(p(t)u')'+q(t)u=f(t,u), t>0\\u(0)=u(+\infty)=0\end{cases}$$ where $\displaystyle\frac{1}{p},\frac{1}{q}\in ...
1
vote
1answer
46 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 ...
0
votes
1answer
24 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 ...
2
votes
1answer
59 views

Sum of the Eigenvalues of a Compact Positive-Definite Linear Operator on a Hilbert Space

Let $ A $ be a compact positive-definite linear operator on a Hilbert space $ \mathcal{H} $. Let $ \{ v_{1},v_{2},\ldots,v_{n} \} $ be an orthonormal $ n $-subset of $ \mathcal{H} $. Let $ \lambda_{1} ...
0
votes
3answers
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) − 3 I know cos(x)= sin (pi/2 -x) but other than that I dont know how to solve this problem
0
votes
0answers
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: ...
1
vote
1answer
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 ...
0
votes
1answer
31 views

$Te_n$ converging to zero

I have the following question in my functional analysis book I dont understand: $X$ is an infinite dimensional Hilbert space with an orthonormal basis $(e_n)$. Show that if $T \in K(X)$, then $Te_n ...
1
vote
1answer
44 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 can`t 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, ...
2
votes
2answers
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.
0
votes
1answer
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 ...
1
vote
1answer
41 views

Small question about strong convergence

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 ...
4
votes
0answers
68 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 ...
1
vote
1answer
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 $\psi \in H^1(0,\pi)$, ...
0
votes
0answers
26 views

Question about function and primitive

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 ...
0
votes
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
0
votes
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 ...
3
votes
1answer
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 ...
-1
votes
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 $a_i$.
1
vote
2answers
45 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 ...
0
votes
1answer
71 views

I dont understand this notation

I`m 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 $B(X)$ ...
0
votes
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: ...
4
votes
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 ...
1
vote
1answer
50 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 ...
0
votes
0answers
21 views

Functional and operator associated to a problem

I have a this functional: associated to the impulsive problem : $$ \begin{cases} -(p(t)u'(t))'=f(t,u(t))\\ u(0)=u(+\infty)=0\\ \Delta(p(t_j)u'(t_j))=h(t_j)I_j(u(t_j)) \end{cases} $$ ...
1
vote
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 ...
1
vote
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)| + ...
0
votes
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 $\|f\|_1=\int |f(x)|dx$
1
vote
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 - ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
2
votes
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 $$ ...
1
vote
0answers
17 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 ...
0
votes
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.