1
vote
0answers
8 views

Convergence of this priori error in FEM?

Problem My attempt I think h is the size of the mesh. C is a constant which probably depends on the size of the mesh, I think. I think the error converges linearly and dependent on the size of ...
1
vote
1answer
23 views

Galerkin Orthogonality in this FEM?

Problem Galerkin orthogonality is but I am not sure if it is in the right form. How can you use this orthogonality here? I think I should expand the last inequality first somehow.
2
votes
1answer
86 views

Uniform convex space

Please I want to know if this space $$H^1_{0,p}([0,+\infty))=\lbrace u, u\in AC([0,+\infty)), u(0)=u(+\infty)=0,\sqrt{p}u'\in L^2\rbrace$$ where $p>0$, $p\in L^1((0,+\infty))$ ...
5
votes
2answers
85 views

What's the spectrum of this operator in $\ell^2$?

Suppose that $\ell^2 = \biggl\{(x_n)_n \in \mathbb{K}^{\mathbb{N}_0} \biggm| \sum_{n=1}^{\infty}|{x_n}^2| < +\infty \biggr\}$ is a Hilbert-space with the inproduct $\langle\cdot,\cdot\rangle_2: ...
0
votes
2answers
26 views

Proof: adjoint map of projection is a projection and …

Let $V$ be a pre hilbert space and $\pi \in \mathrm{End}(V)$. Show: the adjoint map $\pi^+$ of a projection (meaning: $\pi^2 = \pi$) is a projection itself. Show then: a projection $\pi$ is ...
0
votes
0answers
22 views

Extension of a bounded operator on manifold

I have a problem, which is quite urgent. I am given a pseudodifferential operator $A$ in the space $L^0_{\rho,\delta}(M)$, where $M$ is a compact manifold. I wish to extend this operator to an ...
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
0answers
44 views

Prove that if $X$ is a Hilbert space, then $B(X)$ is not a Hilbert space

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. I`m quite stuck and I would love to understand this ...
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 $B(X)$ ...
1
vote
1answer
42 views

Fredholm operators in Hilbert spaces

Suppose $T_r$ and $T_l$ are the left and the right translations in $l_2$. $T_l$ maps $(x_1,x_2,x_3,...)$ to $(x_2,x_3,x_4,...)$, $T_r$ maps $(x_1,x_2,x_3,...)$ to $(0,x_1,x_2,...)$. It can be easily ...
0
votes
1answer
14 views

Image and Kernel of a certain bounded operator

Consider and Hilbert Space $X$, $T\in B(X)$ and a scalar $\mu$ s.t. $|\mu|=||T||$ By a simple argument I deduced that $\ker(\mu I- T)=\ker(\bar\mu I-T^*$) where $*$ denotes the adjoint. I am then ...
3
votes
1answer
83 views

Fourier series to calculate infinite series

I try to show that $\sum_{i=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ using Fourier series and $f(x) = x$ on $L^2_{\mathbb{C}}[-\pi, \pi]$, with basis $e_n(x) = \frac{1}{\sqrt{2\pi}}e^{inx}$. I ...
1
vote
0answers
65 views

The Hilbert space $\mathcal{H}_\eta$ and unitary correspondence with $L^2[a,b]$

The question I have is related to a problem in Stein and Shakarchi's Real Analysis, Chapter 4. The problem Let $\eta(t)$ be a fixed strictly positive continuous function $[a,b]$. Define ...
2
votes
1answer
54 views

Question on a derivative on a Hilbert space

I have this functional $J(u)=\frac12 \|u\|^2+\int_0^1 F(t,Ku(t))dt$ where $F(t,u)=\int_0^u f(t,\xi) d\xi$,$\displaystyle Ku(t)=\int_0^1 G(t,s)u(s) ds$ with $G(t,s)=\begin{cases} s(1-t),&0\leq s ...
0
votes
0answers
23 views

Palais-Smale condition

I have this function: $$\tilde{f}(x)=f(x)+p(||x||)(x_0,x)$$ where $p\in C^2([0,\infty),\mathbb{R}) $ satisfy $0\leq p\leq 1,|p'(t)|\leq \frac{4}{\delta},$ and $$ p(t)= \begin{cases} 1& t\in ...
0
votes
1answer
33 views

Question id a derivative on a Hilbert space

On a Hilbert space $H$; i have this function: $\tilde{f}(x)=f(x)+p(||x||)(x_0,x)$ where $x_0\in H, p\in C^2([0,\infty),\mathbb{R}),f\in C^2(H,\mathbb{R})$ i want to caculate $\tilde{f}', ...
2
votes
2answers
79 views

Derivative on Hilbert space

Please, on a Hilbert space what is the derivative of $\displaystyle\frac{x}{||x||}$ ? I know that it's equal to $\displaystyle \frac{1}{||x||}-\frac{\langle x,\cdot\rangle}{||x||^3} x$ but can I ...
0
votes
1answer
25 views

Check this is a hilbert norm: $ \ell^2 $ with norm $\| \cdot \| := \| \cdot \|_{\ell^2} + \| \cdot \|_{\ell^p}$

Clearly $ p \geq 2 $ so it gains sense calculating the $\ell^p $-norm. According to my calculation this norm is equivalent to the $\ell^2 $ norm, in fact given a cauchy sequence w.r.t $\| \cdot \| $ ...
0
votes
0answers
66 views

Reproducing kernel Hilbert space and finding orthonormal basis from kernel function.

I was trying to extract an orthonormal basis of a reproducing kernel Hilbert space from the expression of the kernel function. I know my answer for finite dimensional reproducing kernel Hilbert space. ...
1
vote
1answer
71 views

$ H $ hilbert space: Hamel dimension of $ H $ = Hilbert dimension of $ H $ $ \Leftrightarrow$ dim $ H $ is finite

Clearly $\Leftarrow $ is a trivial trivial application of G-Schmidt algorithm. I've experienced some trouble in proving the other direction. I focused my self on the fact that span($ A $)=$ H $ (it ...
0
votes
0answers
69 views

Verify that the operator $T$ defined by $T( \varphi _{k})=\frac{1}{k}\varphi _{k+1}$ is compact, but has no eigenvectors.

Let $H$ be a Hilbert space with basis $\left \{ \varphi _{k} \right \}_{k=1}^{\infty }$ .Verify that the operator $T$ defined by $$T( \varphi _{k})=\frac{1}{k}\varphi _{k+1}$$ is compact, but has no ...
0
votes
1answer
45 views

showing that a sequence is converging.

suppose $\left \{ T_{k} \right \}$ is a collection of bounded operators on Hilbert space $H$ ,with $\left \| T_{k} \right \|\leq 1$ for all $k$ .suppose also that $$T_{k}T_{j}^{*}=T_{k}^{*}T_{j}=0 ...
2
votes
1answer
178 views

Inverse of positive operators

Does anyone know how to show this? Let $H$ be a Hilbert space and $A$, $B$ bounded positive operators defined on $H$ such that $A^{-1}: H \rightarrow H$ exists and hence bounded and $A \leq B$. ...
2
votes
1answer
100 views

Eigenvalues and adjoint of operator $T(x_k)_{k=1}^{\infty} = (x_{2k})_{k=1}^{\infty}$

Let $T$: $l^2 \rightarrow l^2$ denote the operator \begin{align} T(x_1,x_2,\dots, x_n,\dots) = (x_2,x_4,\dots,x_{2n},\dots). \end{align} There are several questions regarding this operator that I need ...
4
votes
1answer
120 views

find a weak solution in an intersection of Sobolev spaces

In using lax-milgram to find a weak solution in an intersection of sobolev spaces the weak solution for $$ -\Delta^2 u = f \in L^2(U)\\ \\ u|_{\partial U}=\Delta u|_{\partial U} = 0 $$ was ...
0
votes
2answers
113 views

Hilbert Spaces are Reflexive

I want to show that all Hilbert spaces are reflexive. I have found the following proof on StackExchange: Hilbert Space is reflexive However, I do not understand it. Essentially, we want to show ...
1
vote
2answers
62 views

Orthogonal completion in nonhilbert spaces [duplicate]

Let $X$ be some Hilbert space. There is the widely known theorem in functional analysis which states that for each closed subspace $H\subset X$ we have $H\bigoplus H^{\perp}=X$. Now we do not suppose ...
0
votes
1answer
34 views

Properties of ON-basis in Hilbert space

Let $H$ be a Hilbert space with an ON-basis $(e_n)_{n=1}^\infty$ and let $A$ be a bounded linear mapping $A:H\to H$ such that $$\sum_{n=1}^\infty\|A(e_n)\|^2<\infty$$ 1: Show that if ...
1
vote
1answer
67 views

Bounded linear mappings in Hilbert space preserve orthogonality?

My question is the title of this thread! Assume we have a bounded, linear mapping $A:H\to H$ where $(H,\langle\cdot,\cdot\rangle)$ is a Hilbert space, and two non-zero elements that are orthogonal, ...
1
vote
1answer
36 views

Show that ON-sequence is a base

I have a Hilbert space $H$ and a base $(e_n)_{n=1}^\infty$ and a ON-sequence $(f_n)_{n=1}^\infty$. Given $$ \sum_{n=1}^\infty ||e_n - f_n||^2 < 1 $$ show that $(f_n)_{n=1}^\infty$ is a base. My ...
0
votes
1answer
36 views

$H$ Hilbert, $\ker L \neq H \Rightarrow (\ker L )^{\perp} \neq \lbrace 0 \rbrace$

If $H$ is a Hilbert space on $\mathbb{C}, L : H \rightarrow \mathbb{C} $ is linear and bounded, $\ker L \neq H $ then $ (\ker L )^{\perp} \neq \lbrace 0 \rbrace.$ It seems like a quite easy ...
2
votes
1answer
40 views

Show that $v \in H^1(\Omega)$ if $v\in C^0(\Omega)$ and $v|_{\Omega_j} \in H^1(\Omega_j)$

Let $\Omega$ be an open set in $\Bbb R^d$. Let $\{\Omega_j\}_{j=1}^{N}$ be a fi nite collection of open disjoint subsets of $\Omega$ such that $\overline\Omega=\cup_{j=1}^{N}\overline\Omega_j$. ...
0
votes
2answers
35 views

Representation of a vector

$(l^2,\|\cdot\|_2)$ is a Hilbert space with scalar product $\langle x,y\rangle=\sum^{\infty}_{k=1}x_ky_k$. How can I show that every vector $x\in l^2$ can be written in a form ...
2
votes
1answer
90 views

Calculating the norm of an infinite vector

I'm reading "Introduction to Hilbert Spaces" by N. Young. Right in the first chapter, after introducing inner products and norms in general linear spaces, it asks to show that the norm of the vector: ...
1
vote
1answer
93 views

$\sum c_k^2<\infty$ then $A=\{\sum_{k=1}^{\infty} a_ke_k :|a_k|\leq c_k \}$ is compact

Let $\{e_k\}_{k=1}^\infty$ be an orthonormal set in a Hilbert space $H$. If $\{c_k\}_{k=1}^\infty$ is a sequence of positive real numbers such that $\sum c_k^2<\infty$, then the set: ...
0
votes
1answer
156 views

Hilbert subspace

Let be $H$ Hilbert space and $M\subset H$. $M=M^{\perp\perp}$ if and only if $M$ subspace of $H$. Does anyone know to prove this?
1
vote
1answer
212 views

Operator self-adjoint

I have this paragraph : "Let M be a Hilbert-Riemannian manifold. $f \in C^2(M,R), p \in K$ is called a nondegenerate critical point, if $d^2 f (p)$ has a bounded inverse. Since $A = d^2 f (p)$ is a ...
1
vote
1answer
99 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 ...
1
vote
1answer
51 views

Orthogonal Projectors

Please, I need help with this proble. Let $(H,\langle\cdot,\cdot\rangle)$ be a Hilbert space and let $V_1,V_2,\ldots,V_N$ closed subspaces, mutually orthogonal of $H$, that is, $v_i\perp v_j$ ...
2
votes
2answers
63 views

little question about linear operators

Let H be a complex Hilbert Space. Let $P \in L(H)$ be an idempotent operator ($P^{2} = P$). Also, let $\parallel P\parallel = 1$. I want to prove that $P$ is an orthogonal operator. I defined $M = ...
5
votes
1answer
162 views

Matrix Representation of Trace Class Operators

Suppose we have a separable Hilbert space (thus with a countable basis) and that represent an operator in matrix form, i.e: $A: H \rightarrow H $$$x \;\rightarrow \sum_{j \in \mathbb{N}}\left(\sum_{k ...
5
votes
2answers
386 views

Matrix Representation of Operators in Infinite Dimensional (Separable) Hilbert Spaces

Suppose we have a separable Hilbert space (thus with a countable basis) and that we to represent an operator in matrix form, i.e: $$A: H \rightarrow H \\ \; \; \; \; \; \;x \;\rightarrow \sum_{j \in ...
7
votes
1answer
99 views

Tight Probability on Hilbert space

I am considering the following problem. Let $(X_j)$ be i.i.d. $N(0,1)$ random variables and $H$ a Hilbert space with orthonormal basis $(e_j)$. Let $$X:=\sum_j \frac{X_j e_j}{j}$$ And for any ...
5
votes
1answer
406 views

Using Lax Milgram to find a weak solution in an intersection of Sobolev spaces

I am trying to prove the existence of a weak solution of the problem: $$ -\Delta^2 u = f \in L^2(U)\\ \\ u|_{\partial U}=\Delta u|_{\partial U} = 0 $$ on the bounded open set $U\subset\mathbb{R}^n$ ...
2
votes
1answer
584 views

PDE weak solution problem

My professor grades really strictly (details). I would be very happy if you could help me with this problem: Let $U \subset R^n$ be a bounded set. Consider $ \Delta^2 u = f$ on $U$ and ...
1
vote
1answer
51 views

Particular series on Hilbert Space

Let $(H, \langle\cdot,\cdot\rangle)$ a Hilbert space and consider a sequence $\{x_n\}_{n\in\mathbb{N}}$ of $H$ such that: $$\langle x_n,x_m\rangle\ =\ \delta_{mn}\ =\ \left\{\begin{array}{ll}1, & ...
5
votes
1answer
325 views

Bounded operator and Compactness problem

Let $H$ be a Hilbert space with orthonormal basis $(e_{n})_{n\in\mathbb{N}}$. Furthermore, let $T\colon H\rightarrow C[a,b]$ be a bounded operator. a) Let $x\in [a,b]$. Show that there is a ...
0
votes
1answer
56 views

Hilbert, Banach and isomorphism

I want to show that if linear mapping $L:B_1\rightarrow B_2$ is isomorphism of Banach space and $\|L(x)\|_{B_1} =\|x\|_{B_2} $ (surjective and isometry) so it consist that $L$ is isomorphism of ...
2
votes
1answer
192 views

Invertible operator norm bound

Let $H$ be a Hilbert space and that $X$ are bounded. Suppose $X$ is self-adjoint. Show that $Y=X+iI$ is invertible and the inverse $Y^{-1}$ has the norm $\lVert Y^{-1} \rVert \le 1$. I can prove $Y$ ...