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

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14 views

Are all linear basis functions a reproducing kernel hilbert space?

Do any linear basis function like for instance linear b-splines form a reproducing kernel hilbert space? is it sufficient for the kernel to be semi-positive definite and have a positive Fourier ...
3
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0answers
52 views

Prove that $\bigcap_n K_n \neq ∅$.

Let $H$ be a Hilbert space. Discuss the validity of the following statement: If ${K_n}$ is a decreasing sequence of nonempty, bounded, closed convex sets in $H$, then $\bigcap_n K_n \neq ∅$. ...
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1answer
36 views

If I want to prove that $M^{\perp}$is a closed

If I want to prove that $M^{\perp}$is a closed Can I say because it is the inverse image of $0$ by continuos function ( projection operator )
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1answer
60 views

How can I prove the following theorem with explanation? please

How can I prove the following theorem with explanation. please For any nonempty subset $M$ of a Hilbert space $H$, the span of $M$ is dense in $H$ if and only if $M^{\perp}=\{0\}$ I read the prove ...
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1answer
19 views

ONB of Hilbert dual $H'$

Let $H$ an arbitrary Hilbert space, $\{ e_i \}_{i \in I}$ ONB of $H$. Is there an ONB $\{ e^j \}_{j \in I}$ of the Hilbert dual $H'$, s.t. $e^j(e_i)=\delta_{ij}$? If so, is $\{e_i \otimes e^j\}_{i,j ...
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1answer
30 views

Proving that this space is not Hilbert.

Consider $E$ the space of all the functions defined on $\Bbb R$ which admit a representation of the form $x(t) = \sum_{r \in \Bbb R}^* c_r e^{irt}$, where $\sum^*$ indicates that only a finite number ...
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1answer
32 views

check if a linear operator is bounded

show that $Tf = f(0)$ is not a bounded linear functional on the space of continuous functions measured with the L2 norm, but it is a bounded linear functional if measured using the uniform norm. ...
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1answer
15 views

Diagonal non-compact operator

Suppose we have an operator $I:l_2 \rightarrow l_2$ which is diagonal but not compact. Does that follow: there exists a constant $C$ such that infinite number of diagonal terms $>C$?
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15 views

Dual of Riesz basis with opt. stab. const. $\lambda_\min$, $\lambda_\max$ has opt. stab. const. $\frac1{\lambda_\max}$ and $\frac1{\lambda_\min}$.

Consider a Hilbert space. Consider a Riesz basis $\phi_k$, $k \in \mathcal{K}$ of this space, where $\mathcal{K}$ is an appropriate set of indices. By definition, the Riesz basis fulfils the ...
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1answer
85 views

Theoretical Basis for Eigenvalue transformation on Bessel's Equation

The method I've been taught for finding all of the eigenvalue solutions to Bessel's operator $$b(f)\equiv f''(x)+\frac{1}{x}f'(x)$$ goes as follows. Let $g(a)=f(\sqrt{\lambda}x)$. Then $$b(g)=\lambda ...
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1answer
27 views

Is it true that $M$ is complete?

If $H$ is a Hilbert space and $M$ is a nonempty,closed, bounded and convex subset(not necessarily a subspace)of $H$, then is it true that $M$ is complete? If it is, then can we use it without proof? I ...
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36 views

resource on integral operators

Can you please suggest for me a good resource on integral operators.These are the specific topics that I am looking for: Bounded linear operators in hilbert space. Compact operators Spectral theory ...
3
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1answer
68 views

Hamiltonian: Derivative

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ and an operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^*$$ Denote for shorthand: ...
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0answers
12 views

Concentration of measure of inner product in Hilbert space?

In the finite dimensional Hilbert space of quantum mechanics (one where all vectors have norm one), is a concentration of measure phenomenon observed with the inner product of any two vectors? That ...
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13 views

Concerning projections and hilbert spaces [duplicate]

Suppose $H$ is a Hilbert space and $L $ a subspace of $H$. LEt $\prod$ be projection onto $L$. How is it possible that we can always find a unique element $\prod y$ in $L$ so that $\langle \prod y, z ...
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3answers
65 views

show that $l^2$ is a Hilbert space

Let $l^2$ be the space of square summable sequences with the inner product $\langle x,y\rangle=\sum_\limits{i=1}^\infty x_iy_i$. (a) show that $l^2$ is H Hilbert space. To show that it's a ...
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1answer
30 views

approximation of $x^2$ in hilbert spaces

use the least squares to find the best linear approximation to $f(x)=x^2$ on [-1,1]. that is find the line $y=a_0+a_1x$ that minimizes $\int_{-1}^1|f(x)-y(x)|^2$ solution I used the theory of ...
7
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2answers
108 views

The spectrum of a self-adjoint operator on $\mathcal l^2$

Let $S$ be the unilateral shift operator on $\mathcal l^2$ (which shifts one place to the right) and $S^*$ its adjoint, the backward shift (which shifts one place to the left). I've been trying to ...
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1answer
44 views

Show that $(S^\perp)^\perp=\overline {\operatorname{span}(S)}$ .

Let $H$ be a Hilbert Space. $S\subseteq H$ be a finite set .Show that $(S^\perp)^\perp=\overline {\operatorname{span} (S)}$ . Now $\operatorname{span}(S)$ is the smallest set which contains $S$ and ...
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1answer
17 views

An orthonormal system is total if and only if

Let $H$ be a Hilbert space over a field $\mathbb K$. Prove that an orthonormal system $\{a_n\}_{n=1}^{\infty}$ in $H$ is total if and only if: $\forall$ $x \in H$, the following holds: ...
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24 views

Hamiltonian: Commutator

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ and an operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^*$$ Denote for shorthand: ...
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3answers
48 views

Selfadjoint Operators: Sesquilinear Form (II)

Given a Hilbert space $\mathcal{H}$. Consider a positive form: $$s:\mathcal{D}\to\mathcal{H}:\quad s(\varphi,\varphi)\geq0$$ Introduce its form space: ...
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1answer
43 views

A nonzero compact operator on a Hilbert space has a nonzero eigenvalue

Let $T:H\to H$ be a compact operator on a Hilbert space $H$, with $T\neq0$. Prove that $\exists c\neq0$ and $x\neq0$ such that $Tx=cx$. I was trying to prove it using the fact that if $T$ is a ...
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23 views

Almost sure closeness of random elements in a Hilbert space

Suppose I have a probability space $(\Omega,\mathscr{F},\mathbb{P})$ and a probability measure $\eta$ on a separable Hilbert space $H$ endowed with the Borel $\sigma$-algebra $\mathscr{B}$ arising ...
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1answer
27 views

Selfadjoint Operators: Sesquilinear Form (I)

Given a Hilbert space $\mathcal{H}$. Consider a dense positive form: $$s:\mathcal{D}\times\mathcal{D}\to\mathbb{C}:\quad s(\varphi,\varphi)\geq0\quad(\overline{\mathcal{D}}=\mathcal{H})$$ Construct ...
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1answer
29 views

What is the dual of $H^{-1}(\Omega)$?

The dual of $H^1_0(\Omega)$ is defined to $H^{-1}(\Omega)$. But what is the dual of $H^{-1}(\Omega)$? Is it $H^1_{0}(\Omega)$? I am solving a problem which requires me to use the dual of ...
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1answer
62 views

Invariant subspaces in a Hilbert space

Can someone please help me to answer the following problem? Let $(e_k)$ be a total orthonormal sequence in a separable Hilbert space $H$ and let $T: H \to H$ be defined at $e_k$ by: $T(e_k) = ...
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1answer
18 views

Prove that $\int_{\mathbb{R}}x^me^{2ax}e^{-x^2/2}=e^{2a^2}\int_{\mathbb{R}}(x+2a)^me^{-x^2/2}$

As the title indicates, I've been trying for quite some time now to prove that $$\int_{\mathbb{R}}x^me^{2ax}e^{-x^2/2}=e^{2a^2}\int_{\mathbb{R}}(x+2a)^me^{-x^2/2}$$ $\forall m \in \mathbb{N}, \forall ...
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1answer
35 views

Prob. 1, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $H$ be a Hilbert space, $M \subset X$ a convex subset, and $(x_n)$ a sequence in $M$ such that $\Vert x_n \Vert \to d$, where $d = \inf_{x \in M} \Vert x \Vert$. How to show that $(x_n)$ converges ...
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26 views

Schrodinger Operator with Finite Discrete Spectrum in $(-\infty, -1]$

I'm reading parts of Reed and Simon's Analysis of Operators and have come across a statement I find puzzling. They say that if $V$ is a bounded function of compact support on $\mathbb{R}^3$ then ...
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1answer
28 views

Help required with question about closed unit ball in Hilberts space and proving the projection formula

I've this question that I intend to prove and any help will be appreciated
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1answer
14 views

properties of orthonormal systems and hilbert spaces [closed]

I need to show (a) $\implies$ (b) For an orthonormal system $\{\phi_i\}_{i=1}^\infty$, and a Hilbert space $H$, the following are equivalent: (a) If $\langle f,\phi_i\rangle=0$ $\forall i$, ...
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2answers
43 views

Polarization Identity: Sesquilinearity

Given a vector space $X$. Construct the forms: $$q_s[x]:=s(x,x)\quad s_q(x,y):=\frac{1}{4}\sum_{\alpha=0\ldots3}i^\alpha q[i^\alpha x+y]$$ It is an identification: ...
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0answers
24 views

Using Bounded Operator sequence Theorem

Let E$\subset L_1$ be a set of fourier series functions $e_n(t)=e^{int}$ for $n \in Z$. What is meant by saying to prove $Ge_n$ is a scalar multiple of $e_n$ and it is continuous? How can we prove it? ...
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9 views

Composition with a projection remain surjective in a neighborhood of the parameter

Let $H$ be an Hilbert space and $\varphi:H\to \mathbb R^m$ a smooth map. It is known that the map $u\mapsto d_u\varphi$ is continuous from $H$ to the space of linear operators $L(H,\mathbb R^m)$. ...
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1answer
30 views

can a LUB be part of an open interval

I am re-reading an old textbook "Introduction to Hilbert spaces and applications" by Lokenath Debnath and Piotr Mikusinski, and there is a proof of a lemma in a chapter about the Lebesgue integral ...
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44 views

Explicit inverse of $\lambda-U$ when $U$ is unitary and $|\lambda|<1$

Let $U$ be a unitary operator on a Hilbert space $\mathcal{H}$. By the spectral theorem, it is known that $\sigma(U)\subseteq \{z\in \mathbb{C}:|z|=1\}$. How can the explicit inverse of $\lambda-U$ be ...
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13 views

Normal Operators: Backtransform

Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(T)\to\mathcal{H}:\quad N^*N=NN^*$$ Construct the operator: $$W:=(1+N^*N)^{-1}:\quad Z:=N\sqrt{W}$$ By a ...
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1answer
7 views

Is a Bessel sequence a frame sequence?

$\mathcal H$ being a Hilbert space, $\{g_k\}_{k \in N}$ is a Bessel sequence if there exsits $B >0$ such that $\forall f \in \mathcal H$, $\sum_{k\in N} |\langle f,g_k\rangle|^2 \leq B \| f \|^2$. ...
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13 views

Show that an ONS $\{\phi_n\}$ is complete in $L^2(I)$ iff $\sum_{n=1}^\infty(\int_{[a,x]}\phi_n)^2=x-a$ for all $x\in I$.

Suppose $I=[a,b]$ is an interval of the line. Show that an ONS $\{\phi_n\}$ is complete in $L^2(I)$ iff $\sum_{n=1}^\infty(\int_{[a,x]}\phi_n)^2=x-a$ for all $x\in I$. My Work: If we suppose ...
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1answer
37 views

Normal Operators: Transform

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(T)\to\mathcal{H}:\quad N^*N=NN^*$$ Construct the operator: $$W:=(1+N^*N)^{-1}:\quad Z:=N\sqrt{W}$$ Then it is ...
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1answer
36 views

Prove that an infinite matrix defines a compact operator on $l^2$.

Let $(a(i))_{i=1}^\infty$ be an absolutely summable sequence, i.e., $\sum_{i=1}^\infty |a(i)|<\infty$, and consider the infinite matrix $$A=\begin{bmatrix} a(1)&a(2)&a(3)&\cdots\\ ...
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1answer
35 views

Prove that $min\{\|x-y\|:y\in M\}=max\{|\langle x,y\rangle|:y\in M^\perp , \|y\|=1\}$

Suppose $M$ is a closed subspace of a Hilbert space $X$. Let $x\in X$. Prove that $min\{\|x-y\|:y\in M\}=max\{|\langle x,y\rangle|:y\in M^\perp , \|y\|=1\}$ My Try: First of all I am confused ...
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44 views

Hilbert Space is not locally compact.

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. Show that Hilbert Space is not locally compact at any point. This is what I understand: ...
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1answer
45 views

A criterion for invertibility of a bounded linear map

Let $T$ be a bounded linear operator from the Hilbert space $H_1$ to a Hilbert space $H_2$. Suppose there exists $\delta>0$ such that $$ \langle T^{*}Tx,x\rangle\ge \delta \|x\|^2 $$ for all ...
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1answer
23 views

Show that $L^2(\Omega, \sigma(X),P)$ is a closed hilbert subspace of $L^2(\Omega, \mathbb{A},P)$ s.th $\sigma(X) \subset \mathbb{A}$

I was self-studying probability theory(conditional expectation). I know that a subspace is $U$ of $V$ is a set $U \subset V$ s.th $\forall x,y \in U$ and $\forall \alpha, \beta \in F$ we have that ...
3
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1answer
35 views

If $T^{*}$ is injective then $T$ is surjective?

If $T$ is a bounded linear map from the Hilbert space $H_1$ to the Hilbert space $H_2$, and $T^{*}$ is injective, then I know that $H_2$ is the closure of the range of $T$. But can I conclude that $T$ ...
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1answer
27 views

Characterization of invertibility of bounded linear operator between Hilbert spaces

Let $T$ be a bounded linear operator from the Hilbert space $H_1$ to a Hilbert space $H_2$. I've shown that the existence of a $\delta>0$ such that $$ \langle T^{*}Tx,x\rangle\ge \delta \|x\|^2 ...
0
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1answer
23 views

Injectivity of normal operators on a Hilbert space

Let $A$ be a bounded normal operator on a Hilbert space $H$. I know that $$ \ker A=(\text{ran} A^{*})^{\perp}. $$ What I've been unsuccesfully trying to prove is that $A$ is injective iff its range is ...
2
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1answer
48 views

A representation of a $ C^{*} $-algebra.

I have a quick question about the representation theory of $ C^{*} $-algebras. A representation of a $ C^{*} $-algebra $ A $ is a $ * $-homomorphism $ \pi: A \to B(\mathcal{H}) $, where $ \mathcal{H} ...