For question involving Hilbert spaces, that is, complete normed spaces whose norm comes from an inner product.

learn more… | top users | synonyms

0
votes
0answers
23 views

Fourier basis of $L^2([-\pi,\pi])$

I have read that the Hilbert space $L^2([-\pi,\pi])$ has a Hilbert basis: $$\{e^{inx}|n\in\Bbb{Z}\}$$ This to me indicates that we can only represent a function $u(x)$ by a Fourier Series iff $u(x)\in ...
2
votes
0answers
18 views

integral of product of three basis functions and Clebsh-Gordan coefficients

Suppose I have an orthonormal basis $\{b_i\}_{i=1}^\infty$ for an $L_2$ space (for example, the $b_i$ could be spherical harmonics on the round sphere with the Euclidean $L_2$ inner product). I want ...
2
votes
1answer
32 views

Showing that there exists a sequence that converges weakly in $H_0^1(\Omega)$.

Proof of lemma $9.7$ in Haim Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations argues as follows: For an element $u \in H_0^1(\Omega)$ we define $D_h u= \frac{u(x+h)-u(x)}...
1
vote
0answers
18 views

Fatou lemma and weak convergence in Hilbert

In a Hilbert space $H$ a sequence $(x_n)_{n\geq0}$ is said to converge weakly to $x$ if $\forall y\in H:\langle y,x_n\rangle\rightarrow\langle y,x\rangle$, the case in which we can easily deduce an ...
0
votes
0answers
33 views

Distance preserving function on hilbert space

It is known that an isometry on B(H) is distance preserving .I am trying to show ,conversely , that if F=R,every distance preserving function f on H( Hilbert space) has the form f(x) = f(0) +Tx for ...
0
votes
1answer
21 views

Conjugate linear isometry of Hilbert operators

Let $H$ and $R$ be Hilbert spaces and consider an operator $T$ in $B(H,R)$. I need to show that there is a unique operator $T^*$ in $B(R,H)$ satisfying $$(Tx│y)_R = (x│ T^* y)_H,$$ $x \in H$, $y \in R$...
1
vote
1answer
38 views

Generalized polar decomposition

Let $x\in B(H)$. We say $(x,v,y)$ is a polar decomposition for $x$ if, $\bullet$ $y$ is positive. $\bullet$ $v$ is a partial isometry with $x=vy$. $\bullet$ Ker$(x)$=Ker$(y)$=Ker($v$) The polar ...
1
vote
1answer
36 views

Let $H$ be a Hilbert space, $V≤H$ be closed, $Q:H→V$ be the orthogonal projection, $(e_n)_{n∈ℕ}$ be an ONB of $H$. Is $(Qe_n)_{n∈ℕ}$ an ONB of $V$?

Let $\mathbb K\in\left\{\mathbb C,\mathbb R\right\}$ $U$ and $H$ be $\mathbb K$-Hilbert spaces $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $\iota:U\to H$ be an embedding and $V:=\iota(U)$ ...
2
votes
1answer
20 views

Orthogonal sequences under A Hillbert space

I know that for two vectors $u,v\in H$ where $H$ is a Hilbert space the definition for orthogonality is $\langle u,v \rangle =0$. is thaat also corret for sequences? What is the definition for ...
2
votes
1answer
49 views

Prove a sequence is bounded under a Hilbert space

Let $T:H\to H$ be defined by $Tx=\sum_{n=1}^\infty \lambda_n \langle x,\varphi_n \rangle \varphi_n$ where $\{\varphi_n\}_{n=1}^\infty$ is an orthogonal sequence and $\{\lambda_n\}_{n=1}^\infty$ is a ...
0
votes
1answer
31 views

Linear Operator on Hilbert Space $l(\mathbb Z)$

Let $A$ be the linear operator on $l(\mathbb Z)$ defined for $u=\{u_k\}_{k \in \mathbb Z}$ as $(Au)_k = \sum_{h=-\infty}^{+\infty}a_{k,h}u_h$ where $a_{k,h}=\frac{1}{(k-h)2}$ for $h \not= k$, and $...
1
vote
0answers
49 views

Linear Operators on $L_2(\mathbb R)$ definfed as Integrals

Let's consider the linear operators on $L_2(\mathbb R)$ $$ T_{\alpha}f(x) = \int_{-\infty}^{+\infty} \frac{e^{-|x-y|^2}}{(1+x^2)^{\alpha}}f(y)dy $$ with ${\alpha} \in [0,1]$. Find ${\alpha}$ such ...
3
votes
0answers
37 views

Square-root of $\iota\iota^\ast$, where $\iota$ is an isometric embedding between Hilbert spaces

Let $U$ and $H$ be Hilbert spaces and $\iota$ be an embedding of $U$ into $H$. Then, $$\pi x:=u\;\;\;\text{for }x\in H\text{ with }x=\iota u+y\text{ for some }u\in U\text{ and }y\in\left(\iota U\right)...
1
vote
0answers
25 views

Minimal uniquely achieved of a funcional in space Hilbert

I have the follow problem : Let be $H$ a Hilbert space and $E \subset H$ a closed space of $H$. Give a $w \in H$ such that $w \not \in E^\perp$, show that $$\inf_{v \in E, |v| = 1}\big<w,v\big> ...
2
votes
1answer
24 views

The point spectrum and residual spectrum of an operator on $l_2$ related to backward shift

I have a problem with the spectrum of this operator: $(Tx)_1 = x_2$ $(Tx)_2 = x_1$ $(Tx)_n = \frac{1}{n}x_{n+1}$ with $n\ge3$ Find the $||T||$, the point spectrum $\sigma_P(T)$ and $\sigma_P(T^{\...
2
votes
2answers
51 views

Show that $\ker(T)=\{\varphi _n\mid\lambda_n\neq 0\}^\perp $

Let $T:H \to H$ be defined as $Tx=\sum_{n=1}^{\infty} \lambda_n \langle x,\varphi _n \rangle \varphi _n$, given that $\{\varphi _n\}_{n=1}^\infty$ is an orthonormal sequence (not necessarily a basis) ...
0
votes
0answers
30 views

Given a special Hilbert space $U_0$, is there a proper superspace $V$ such that the inclusion $\iota:U_0\to V$ is Hilbert-Schmidt?

Let $U$ be a Hilbert space $Q\in\mathfrak L(U)$ be nonnegative and symmetric $U_0:=Q^{1/2}U$ be equipped with $$\langle u,v\rangle_{U_0}:=\langle Q^{-1/2}u,Q^{-1/2}v\rangle_U\;\;\;\text{for }u,v\in ...
1
vote
1answer
44 views

Show that a sequence does not converge

I have a very similar question to this and so I'll change my letters to match: Let $\{\varphi_n\}^∞_{n=1}$ be an orthonormal sequence (not necessarily a basis) in a Hilbert space. Let $\{λ_n\}^\...
2
votes
3answers
168 views

Convergence under a Hilbert space

Let $\{\varphi_n\}_{n=1}^\infty$ be an orthonormal sequence (not necessarily a basis) in a Hilbert space. Let $\{\lambda_n\}_{n=1}^\infty$ be a sequence of numbers Define $T:H\to H$ by $Tx= \sum_{n=...
0
votes
1answer
34 views

If a positive operator $y$ has the same kernel as $cy$, what can we conclude about the kernel of $c$?

Let us consider the equation $x=cy$ in $B(H)$. Assume that: $y$ is a positive operator. $x$ and $y$ have the same null space. Ker($y$) is contained in Ker($c$). Can we conclude that Ker($y$)=...
1
vote
1answer
36 views

$L^2$ convergence of partial sum of a sequence of functions: $\sum_{k=1}^n\frac{g^k(x)}{k}$

Let $g:\mathbb{R}\to \mathbb{C}$ be an $L^2$ function such that $|g(x)|\leq\epsilon<1, $ for every $x\in\mathbb{R}$, and $g(x) = O\left(\frac{1}{x}\right)$. I want to know if $h_n(x):= \sum_{k=1}^n ...
1
vote
1answer
33 views

Can we find a concrete representation of $\iota\iota^\ast y$, if $\iota$ is a Hilbert-Schmidt embedding between Hilbert spaces?

Let $U$ and $H$ be real Hilbert spaces $\iota:U\to H$ be a Hilbert-Schmidt embedding $Q:=\iota\iota^\ast$ Can we find a concrete representation of $Qy$ for some $y\in H$? By Riesz' ...
0
votes
0answers
19 views

If $U_0,V$ are Hilbert spaces, $(e_n)$ is an ONB of $U_0$ and $ι:U_0→V$ is an embedding, can we complete $(ιe_n)$ to an ONB of $V$?

Let $U$ and $H$ be Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric $U_0:=Q^{1/2}U$ be equipped with $$\langle u,v\rangle_{U_0}:=\langle Q^{-1/2}u,Q^{-1/2}v\rangle_U\;\;\;\text{for }...
0
votes
0answers
27 views

Proof Verification: Separable Hilbert Space has a Countable Orthonormal Basis?

I was browsing proofs of this involving the Gram-Schmidt process when the following occurred to me. I'd appreciate feedback. If $H$ is a separable Hilbert space then it has a countable dense subset $...
4
votes
1answer
33 views

Is the Null Space of an linear operator the same with the Null Space of its associated hermitian?

Let A be a bounded linear operator on $H$ where $H$ is a (not necessary I think, but in my case separable) Hilbert space. Then, the question: is its null space the same as the null space of the ...
1
vote
1answer
48 views

Is the completion of $C_0^\infty(\mathbb{R}^n)$ with respect to $\int_{\mathbb{R}^n}| \nabla \varphi|^2dx$ contained in $L^2(\mathbb{R}^n)$?

Equip $C_0^\infty(\mathbb{R}^n)$ with the norm $$ \|\varphi\|^2_1 := \int_{\mathbb{R}^n}| \nabla \varphi|^2dx.$$ Indeed, $\| \cdot \|_1$ is a norm on $C_0^\infty(\mathbb{R}^n)$ because any constant ...
0
votes
0answers
39 views

Norm of an operator in a Hilbert space

Let $T\neq 0, \neq I$ be a linear operator of a Hilbert space such that $T \circ T = T $. Show that $\|T\|=\|I-T\|$. Anyone has an idea ? I just proved that $\|I-T\| \leq 1 + \|T\|$ but it is not ...
1
vote
2answers
27 views

For a densely defined symmetric operator $A$, is $A^2$ also densely defined?

Let $A : D(A) \to H$ be a possibly unbounded, densely defined symmetric operator on a Hilbert space $H$ ($A$ being symmetric means that $(\varphi, A\psi) = (A\varphi, \psi)$ for all $\varphi, \psi \in ...
0
votes
1answer
41 views

A factorization for operators

Let $a$ be an arbitrary operator in $B(H)$ and $b$ be a positive operator in $B(H)$. Assume $a$ and $b$ have the same null space and there exists an operator $u\in B(H)$ with $a=ub$. Q) Can we ...
0
votes
0answers
34 views

I don't understand how the adjoint operator is used in a book that I'm reading

I'm reading Stochastic Differential Equations in Infinite Dimensions and don't understand what the authors do in Chapter 2.3.1. Let me introduce the necessary objects: Let $K$ and $H$ be real ...
0
votes
0answers
23 views

Is there a o.g basis of $H^1(\Omega)$ that is o.n in $L^2(\Omega)$?

Let $\Omega$ be a bounded smooth domain, and define $H^1(\Omega)$ with the usual norm involving the function and its gradient. I am wanting to know if $H^1(\Omega)$ possesses a basis $b_j$ such that $...
1
vote
2answers
40 views

Minimizing an integral — Hilbert space

Find the real values of $a, b$ which minimize $$\int_1^{\infty} \left| \frac{1}{x^2} - a \frac{1}{x^3} - b\frac{1}{x^4}\right|^2 \; dx.$$ Hint : Work in an appropriate Hilbert space. Here is why I ...
0
votes
0answers
26 views

I want to calulate the range of an operator $S$ which maps $L^{2}$ into $H^{1}(\Omega)$?

I have an equation $s(\lambda,\mu)=l(\mu)$, where s(.,.) is a symetric positive definite bilinear form in $L^{2}(\Omega)$, and $l(.)$ is in $H^{1}(\Omega)$. I want to show that the range of the ...
1
vote
0answers
38 views

If $Q$ is a trace class operator on $U$, then each bounded, linear operator from $U$ to $H$ is a Hilbert-Schmidt operator from $Q^{1/2}U$ to $H$

Let $U$ and $H$ be Hilbert spaces $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ with finite trace $U_0:=Q^{1/2}U$ $L$ be a bounded, linear operator from $U_0$ to $H$ I ...
1
vote
1answer
31 views

How to show that this integral operator is bounded?

Consider the integral operator $T : C([0,1])\to C([0,1])$ given by $$Tf(t)=\int_0^1 K(t,\tau)f(\tau)d\tau.$$ I'm solving one exercise which is to show this operator is bounded. The exercise is from ...
1
vote
2answers
27 views

Showing $A_{ij} = (Ae_i, e_j)$ for matrix $A$ of complex linear operator $\mathbb{C}^n \to \mathbb{C}^n$ and orthonormal basis $(e_i)_{i=1}^{n}$

I'm sure this is a simple question, but I get stuck in the algebra when I try to prove it from definition. Suppose $A$ is an $n \times n$ matrix representing a complex linear operator $\mathbb{C}^n \...
2
votes
1answer
36 views

Operator with norm

I got the following problem to solve: Let $H$ Hilbert space and $T: H \to H$ a bounded positive operator, i.e. \begin{align*} \langle x, T x \rangle \geq 0 & & \text{for all } x \in H. \end{...
1
vote
1answer
31 views

Bounded operators on inner products on Hilbert space

If we have a Hilbert space $H$ with inner product $( \cdot | \cdot)$, and let $( \cdot| \cdot)_1 $ be another inner product on $H$ such that $(x | x)_1 \leq (x | x)$ for every $x \in H$. I was trying ...
1
vote
1answer
21 views

Convergence of unitary products on a Hilbert space

First: I'm sorry for the basic question--I can move it to Math SE if necessary... Let $X$ be a Hilbert space and suppose $\{U_k\}_k$ is a sequence of unitary operators on $X$. Let $\|\cdot\|$ be the ...
1
vote
1answer
37 views

Proof about orthogonality of columns of a matrix

Consider a matrix $A \in \mathbb{R}^{n \times n}$ and the canonical inner product in $\mathbb{R}^{n}$. Show that if the rows of A form an orthogonal set, the same happens with the columns. So what ...
0
votes
0answers
22 views

Hausdorff-Quotient: Embedding

Problem Given a uniform space $\Omega$. (Exemplary Topological Vector Space!) Consider a dense subspace: $$\iota:\mathcal{D}\hookrightarrow\Omega:\quad\overline{\iota\mathcal{D}}=\Omega$$ Regard ...
2
votes
1answer
31 views

Convexity of Hilbert cube [closed]

I am trying to show that the Hilbert cube $\{ x_n \in l^2(\mathbb{N}) \mid x_n \in [0, \frac{1}{n}] \ \forall n \in \mathbb{N} \}$ is convex and (norm)-compact.
0
votes
1answer
54 views

Cardinality of a Hilbert space

I have seen the theorem about the cardinality of orthonormal basis of a Hilbert space. I wonder if we have a Hilbert space $H$ with an orthonormal basis having cardinality of the continuum, then what ...
1
vote
1answer
56 views

Bounded operator on Hilbert space

Let $H$ is a Hilbert space. If $T\in B(H)$ show that $T+T^*\ge 0$ iff $T+I$ is invertible in $B(H)$ with $\|(T-I)(T+I)^{-1}\|\le 1$. (Hint is $T+T^*\ge 0$ iff $\|(T+I)x\|\ge \|x\|\ $ and $\|(T+I)x\...
0
votes
1answer
21 views

Generalized Poincaré Inequality on H1 proof

let's see if someone can help me with this proof. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. And let $L^2\left(\Omega\right)$ be the space of equivalence classes of square integrable ...
2
votes
1answer
59 views

Generalized Poincaré Inequality on H1 proof.

let's see if someone can help me with this proof. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. And let $L^2\left(\Omega\right)$ be the space of equivalence classes of square integrable ...
2
votes
0answers
21 views

Norm of infinite dimensional Hilbert space to calculate difference between string lengths

I am trying to wrap my head around Proposition 13, last para, page 1049 in this paper. The authors are trying to prove certain properties of string edit distance (defined at the start of Section of 6....
1
vote
1answer
20 views

A bounded linear functional on a Hilbert space that is a Hahn-Banach extension of one on a subspace

Let $M$ be a closed linear subspace of a Hilbert space $H$ and $g\in M*$(all bounded linear functional on $M$). Let $\pi$ be the orthogonal projection of H onto M, then $f=g\circ\pi$ is the only Hahn-...
2
votes
1answer
79 views

Exercise 2 , chapter 5 , Stein & Shakarchi real analysis

Consider the Mellin transform defined initially for continuous function $f$ of compact support in $R^+=${$t\in R:t>0$} and $x\in R$ by $Mf(x)=\int_0^\infty f(t)t^{ix-1}dt$ Prove that ($2\pi$)$^{-...
0
votes
1answer
15 views

Separable infinite-dimension Hilbert space and its subspaces

Suppose $H$ is any separable infinite-dimensional Hilbert space. Then $H$ has family of closed subspaces $\big\{ E_t :~ t \in [0,1]\big\}$ such that $E_s$ is a strict subspace of $E_t$ for all $0 \leq ...