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

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3
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2answers
42 views

Show $ \langle Tx,x \rangle \in \mathbb R$ for all $x \in H$ implies $T$ is self-adjoint

Show that a linear operator $T: H \rightarrow H$ is self adjoint if and only if $\langle Tx, x \rangle \in \mathbb R$ for all $x \in H$. You may use that the equality that for all $x,y \in H$ ...
0
votes
0answers
29 views

Is there a pseudocontractive mapping that is not strictly pseudocontractive?

Given a Hilbert space $H$, a mapping $T:H\rightarrow H$ is said to be pseudocontractive if $$\|Tx-Ty\|^2\leq \|x-y\|^2+\|(x-Tx)-(y-Ty)\|^2\,\,\, \forall x,y\in H,$$ and it is strictly ...
1
vote
1answer
13 views

A rescaled inner product inequality

I was wondering if the following inequality is true: Let $\xi_1,...,\xi_n$ be vectors in a Hilbert space $H$ and let $x_{i,j}$ be complex numbers such that $\prod x_{i,j}$ is real and $$\prod ...
2
votes
1answer
24 views

A question on Isometry between the orthogonal subspaces of Hilbert spaces

I was reviewing my class-notes on Functional analysis and the professor had mentioned that given a closed proper subspace $U$ of an hilbert space $\mathcal{H}$, $\exists $ a closed subspace ...
6
votes
1answer
82 views

$\sin$ and $\cos$ are the basis of what space?

When learning Fourier expansions, we learn that $\{\sin(mx), \cos(mx)\}_{m \in \Bbb N}$ is an orthonormal basis for our space and thus we can expand functions in it. My question is what space is this ...
0
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0answers
26 views

Cauchy sequence in reproducing kernel Hilbert space

Consider a positive definite kernel $K:\mathbb N\times \mathbb N\rightarrow \mathbb R$. Denote the unique RKHS associated with $K$ by $\mathcal H_K$. The RKHS $\mathcal H_K$ consists of \begin{align} ...
1
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0answers
63 views

$P$ and $Q$ are unitarily equivalent iff dimensions of ranges and kernels are the same

Two projections $P,Q$ are unitarily equivalent if and only if $$dim(randP)=dim(ranQ)$$ $$dim(kerP)=dim(kerQ)$$ How can we show this? One directionn seems easy: If $P$ and $Q$ are unitarily eqv, ...
1
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1answer
78 views

How do I prove a differential operator has no purely imaginary eigenvalues?

Anyone who has taken a course in linear algebra knows how to prove the eigenvalues of a self-adjoint operator are real or the eigenvalues of a skew-self-adjoint operator are purely imaginary. This is ...
5
votes
2answers
74 views

Prob. 8, Sec. 3.5 in Erwin Kreyszig's Introductory Functoinal Anlaysis With Applications

Erwin Kreyszig's Introductory Functoinal Anlaysis With Applications Prob. 8, Sec. 3.5 $\DeclareMathOperator{\span}{span}$Let $(e_k)$ be an orthonormal sequence in a Hilbert space $H$, and let $M = ...
0
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2answers
49 views

A question about Fourier coefficients.

Is it true that the sequences $ (A_{n})_{n \in \Bbb{N}} = (0)_{n \in \Bbb{N}} $ and $ (B_{n})_{n \in \Bbb{N}} = \left( \dfrac{1}{\sqrt{n}} \right)_{n \in \Bbb{N}} $ are the Fourier coefficients of ...
0
votes
0answers
9 views

Existence of Schauder base for given operator

Suppose $A: l_2 \rightarrow l_2$ is a finite-rank linear bounded operator of dimension $k$. Is it true that there exists a Schauder orthonormal base for which only first $k$ columns will be nonzero ...
0
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1answer
26 views

is $\langle\lim_{n\to \infty}u_n,g\rangle = \lim_{n\to\infty} \langle u,g\rangle $ valid for bounded linear operators?

Suppose M is any linear manifold in H. H is a hilbert space. Define the orthogonal complement of M to be $$M' =\{f \in H | \langle f,g\rangle= 0 ,\forall g\in M\}.$$ To see that M' is a closed ...
0
votes
0answers
16 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
votes
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 $\bigcap_n K_n \neq ∅$. ...
2
votes
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 )
0
votes
1answer
61 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 ...
1
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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 ...
2
votes
1answer
34 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 ...
1
vote
1answer
38 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. ...
0
votes
1answer
20 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 $>C$?
1
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0answers
19 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 ...
10
votes
1answer
99 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 ...
0
votes
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 ...
3
votes
1answer
71 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: ...
1
vote
0answers
17 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 ...
0
votes
0answers
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 ...
0
votes
3answers
78 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 ...
0
votes
1answer
32 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
votes
2answers
112 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 ...
0
votes
1answer
48 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 ...
0
votes
1answer
19 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: ...
1
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0answers
25 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: ...
0
votes
3answers
55 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: ...
0
votes
1answer
51 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 ...
0
votes
0answers
24 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 ...
0
votes
1answer
31 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 ...
0
votes
1answer
36 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 ...
1
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1answer
64 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) = ...
0
votes
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 ...
0
votes
1answer
47 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 ...
4
votes
0answers
85 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 ...
2
votes
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
-1
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1answer
17 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 $\langle f,\phi_i\rangle=0$ $\forall i$, ...
0
votes
2answers
60 views

Sesquilinear Forms: Parallelogram

Given a Hilbert space $\mathcal{H}$. Consider a quadratic form: $$q:\mathcal{H}\to\mathbb{C}:\quad q[\lambda\varphi]=|\lambda|^2q[\varphi]$$ Suppose one has: ...
0
votes
0answers
25 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? ...
0
votes
0answers
11 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 $L(H,\mathbb R^m)$. ...
1
vote
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 ...
1
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0answers
50 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 ...
0
votes
1answer
50 views

Normal Operators: Transform (II)

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ Construct the operator: $$Q:=(1+N^*N)^{-1}:\quad Z:=N\sqrt{Q}$$ By the previous ...
1
vote
1answer
11 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 $\forall f \in \mathcal H$, $\sum_{k\in N} |\langle f,g_k\rangle|^2 \leq B \| f \|^2$. ...