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

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$A^{\perp\perp}=A$ for a closed subspace $A$ [duplicate]

This question is essentially the same as double Orthogonal complement is equal to topological closure, but I have fewer "known" results. (Specifically, the linked Q/A proves that ...
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50 views

To show Banach Operator Norm's equivalence

I am thinking the sentence which may be about the Banach algebra function: Let $A \in \mathcal{B}(H)$ where $H$ is Hilbert. \begin{equation} \| A \| = \sup_{ \| u \|, \| v \| \leq 1} | \langle Au, ...
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30 views

Is the integral form of the polarisation identity useful for anything?

It is well-known that the polarisation identity for real vector spaces is $$ \langle a,b \rangle =\frac{1}{2}\sum_{k=0}^1 (-1)^k\lVert a+(-1)^k b \rVert^2, $$ and the complex generalisation is $$ ...
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2answers
51 views

Consider the Hilbert product space $X\times X$

Consider the Hilbert product space $X\times X$. In $X\times X$ define the closed convex 'diagonal' set by $$D={(x,x):x\in X}$$ Obtain a formula for projection $P_D$ and rigorously prove it. I really ...
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43 views

Complete orthonormal system of eigenfunctions for trace-class nonnegative operator on a Hilbert space

In Da Prato/Zabczyk's book "Stochastic equations in inifinite dimension" I stumbled over the following paragraph: Let $Q$ be a trace class nonnegative operator on a Hilbert space $U$. [...] Note that ...
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123 views

Trace Class: Relativeness

Given a Hilbert space $\mathcal{H}$. Consider a selfadjoint: $$H\in\mathcal{B}(\mathcal{H}):\quad H=H^*$$ Denote trace class: ...
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15 views

Some example about orthonormal continuous bases

I have an assignment in where i need to prove if a given continuous base is orthonormal and complete. I have the theory but no examples as a starting point. $$\phi_n (k,x) = \Bigg\{ ...
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1answer
17 views

An identity regarding linear operators and their adjoint between Hilbert spaces

I came upon a trivial-seeming claim that I can't prove myself: Let $H_1$, $H_2$ and $H_3$ be finite-dimensional Hilbert spaces, let $A:H_1\rightarrow H_2$ and $B,C:H_3\rightarrow H_1$ be linear ...
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1answer
43 views

Orthogonality in Hilbert Spaces

For the sake of concreteness, let's say that our Hilbert space is the beloved $\mathscr L^2(\Bbb R)$. Suppose that we have $\psi,\phi\in\mathscr L^2(\Bbb R)$, what's the intuitive meaning to a ...
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1answer
105 views

Why are inner product spaces only defined on $\Bbb R$ or $\Bbb C$?

A vector space $V$ makes sense over any field $F$, or even a division ring. So why does adding an inner product suddenly not make sense without taking the $F=\Bbb R$ or $\Bbb C$? What are the primary ...
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27 views

Stability of least isolated eigenvalue under positive perturbation

This would be a useful theorem. Have you seen it anywhere? $\mathbf{Theorem:}$ Suppose a self-adjoint operator $H_0$ on a Hilbert space has a simple isolated least eigenvalue $0$ with separation ...
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23 views

Orthogonality in $H^1(T)$ with the inner product $(\cdot,\cdot)_{L^2(T)}$

Take the Sobolev space $$H^1(T)=\left\{f \in L^2(T) ~|~ f' \in L^2(T)\right\}$$ where $T$ is the 1-torus (that is a circle) and $f'$ the weak derivative. Take then a function $v\in H^1(T)$ and ...
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1answer
16 views

rad(T)=||T|| for non-normal T

It is well-known that for normal bounded operators $T$ on a Hilbert space one has $\mathrm{rad}(T)=\|T\|$ (where rad is the spectral radius). Are there any sufficient conditions under which a ...
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1answer
50 views

Proof for index of pair of operators

Let $P$ and $Q$ be a pair of orthogonal projections on a separable Hilbert space $\mathcal{H}$, such that $P-Q$ is compact and $(P-Q)^{2n-1}$ is trace class for some $n\in\mathbb{N}_{\geq0}$. Claim: ...
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1answer
112 views

Idempotent projection operators in a Hilbert space

Let $H$ be a Hilbert-space and $S$ be a sub(Hilbert)space such that: $$H = S \oplus S^\perp$$ Then the projection operators are defined as: $$P_S: H\to S; x = u + v \mapsto u \quad\quad P_{S\perp}: ...
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178 views

Is $\int |x\rangle \langle x|dx$ Mathematical?

I am enrolling in a Quantum Mechanics class. As we all know, the formulation of the basic ideas from QM relies heavily on the notion of Hilbert Space. I decide to take the course since it might help ...
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1answer
260 views

Proving the closed unit ball of a Hilbert space is weakly sequentially compact

I bumped into this statement in Hofer-Zehnder in the middle of proving a Hamiltonian field always has a periodic orbit over a level set of the hamiltonian if that set is a regular compact and strictly ...
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2answers
30 views

Prove product of projections is bounded

Let $P$ and $Q$ be two orthogonal projections (so that means they are linear, idempotent, and self adjoint) such that $P-Q$ is a compact operator (so that means the closure of $(P-Q)(B_1(0))$ is ...
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1answer
73 views

Closed unit ball of Hilbert space: sequentially compact in weak topology?

Following the proof of the existence theorem in chapter 1 of Hofer-Zehnder, Symplectic Invariants and Hamiltonian Dynamics, I find: We have used the well-known fact that the closed unit ball of a ...
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1answer
50 views

Connection between the Fourier transform of a measure and the support of the measure

Let $H$ be a separable real Hilbert space, $\mu$ a complex finite Borel measure on it with Fourier transform $\hat \mu$, and $V$ a finite dimensional subspace. Let $P : H \to V$ be the orthogonal ...
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52 views

Continuity of the Fourier transform of a measure

If $\mu$ is a complex finite Borel measure on a separable real Hilbert space $H$ then $$x \mapsto \hat \mu (x) = \int \limits _H \Bbb e ^{\Bbb i \langle x, y \rangle } \Bbb d \mu _{(y)}$$ is ...
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102 views

If the Fourier transform of a measure is zero then the measure is zero

If $\mu$ is a complex finite Borel measure on a separable real Hilbert space $H$ be such that $$\hat \mu (x) = \int \limits _H \Bbb e ^{\Bbb i \langle x, y \rangle} \Bbb d \mu _{(y)} = 0, \ \forall ...
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59 views

Operator realizations of algebras

I want to realize the algebra $A_q(\tilde{S}^{n-1})$ as introduced in the acticle of Dijkhuizen and Noumi (http://arxiv.org/pdf/q-alg/9605017v1.pdf) as bounded operators on a Hilbert space $H$. Can ...
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1answer
57 views

How to rigorously understand continuous bases?

In Quantum Mechanics it is quite common to see the idea of a continuous basis of a Hilbert space. In truth if $\mathcal{H}$ is the state space of a quantum system and if $X : U\subset \mathcal{H}\to ...
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89 views

Prove that $A\geq I$ implies that $A$ is invertible.

Here's the question: Let $A$ be a positive operator on a (possibly infinite dimensional) Hilbert space. Let $I$ denote the identity operator. Suppose that $A \geq I$, which is to say that $A - ...
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36 views

Show that $l_{2}(J)$ is Hilbert Space for Countably Infinite Set?

The inner product is \begin{equation*} \langle u, v \rangle = \sum\limits_{j \in J} u_{j} \overline{v_{j}} \end{equation*} where $u,v$ are vectors and $J$ is the countably infinite set $J = ...
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1answer
45 views

If $(e_1,\ldots)$ is an orthonormal basis and $x$ is orthogonal to the $(e_i)$, must $x$ be $0$?

Let $H,\langle\rangle$ be a pre-Hilbert space and $(e_1,\ldots,e_n,\ldots)$ an orthonormal basis of $H$. Suppose that there is some $x$ such that $\forall i, \langle x,e_i\rangle=0$ Must ...
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1answer
27 views

How do we define the Hermitian adjoint of $u$ in the case where the domain and codomain of $u$ differ?

According to wikipedia: If $X$ and $Y$ are Hilbert spaces and $u : X \rightarrow Y$ is a linear map, then the transpose of $u$... and the Hermitian adjoint of $u$... are related. I don't get ...
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102 views

Orthogonal complement for infinite dimensional and open vector spaces.

Given a vector space $V$ and a subspace $U$, the orthogonal complement $U^{\bot}$ have the following properties: $U^{\bot}$ is itself a subspace of V. $U \cap U^{\bot} = \{0\}$. $(U^{\bot})^{\bot} ...
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20 views

Zeros of functions constituting a Riesz-basis for the Paley-Wiener space

I have a short question which first requires some slightly elaborate definitions: Let $(e_n)$ be a Riesz-basis for a Hilbert space $\mathcal{H}$ with biorthogonal basis $(g_n)$, i.e. $\langle e_m, ...
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22 views

Orthogonal projection by standard norm of and a non-standard norm in $\mathbb{R}^2$

Suppose a closed convex set $S\subset \mathbb{R}^2$ is given by by the convex hull of $(0,1) (-1,1), (-1,0), (1,0)$ and a continuous, convex and decreasing curve $F$ linking $(1,0)$ and $(0,1)$, ...
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65 views

Proving Toeplitz matrix defines bounded operator on $ l^2 $

I should first mention this: I have asked this question previously but I only got a partial answer that does not suit the actual assumptions but only the related ones, it reads as follows: Define ...
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3answers
43 views

Prove that functions $\phi$ that are zero except on some finite subset of $A$ are dense in $\ell^2(A)$

In his book "Real and Complex Analysis", chapter "Elementary Hilbert Space Theory", Walter Rudin introduces the following theorem: Let $\ell^2(A)=\{\phi: \longrightarrow \mathbb{C}\; | \; \sum_{a\in ...
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1answer
30 views

Prove that $|x+y|^2-|x-y|^2+i|x+iy|^2-i|x-iy|^2=4x\overline{y}$

I am trying to prove that the following equation holds for every complex number $x,y$: $|x+y|^2-|x-y|^2+i|x+iy|^2-i|x-iy|^2=4x\overline{y}$, where $i=\sqrt{-1}$ and $\overline{y}$ is the complex ...
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27 views

Double series of linear operators

Does anybody know something about double series of linear operators on a Hilbert space? Specifically, given an infinite-dimensional, complex separable Hilbert space $\mathcal{H}$, with an orthonormal ...
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1answer
28 views

Will the problem right?

Let $X$ be a normed space and $Y_1$ , $Y_2$ complete spaces and exist two injective linear application $f_1:X \to Y_1$, $f_2:X\to Y_2$ such that $\text{closing } f_1(X)=Y_1$, $\text{closing } ...
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48 views

all possible inner products in $\mathbb R^2$

Suppose $\langle., .\rangle: \mathbb R^2\times \mathbb R^2\to \mathbb R$ is an inner product. What would be all possible function forms of the inner products, i.e. would all of them have the forms ...
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Progressed : Convergence problem in Hilbert Space and necessity of inner product

******** PROGRESS : so thanks to Ian's great comment I can get by the proof and that completeness is necessary but I need to know does this hold for general Banach spaces that are not Hilbert spaces? ...
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1answer
75 views

Quantum Mechanics state space

In Quantum Mechanics one often deals with wavefunctions of particles. In that case, it is natural to consider as the space of states the space $L^2(\mathbb{R}^3)$. On the other hand, on the book I'm ...
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25 views

Completeness property in signal analysis

Why completeness is an important property for signal analysis such as Fourier? What if we don't have a such a property?Many books discuss that the vector should not have a hole to complete.what is ...
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39 views

Hilbert space from signal processing view! [closed]

I am electrical engineer and not so much deep into math. I have one question that i think some of you may enlighten me up. The concept Hilbert space; I know the idea is related to Fourier ...
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25 views

Simple example of a not strongly continuous operator on a Hilbert space?

Let $\cal H$ be a Hilbert space. Let $U(t)$ with $t\in \mathbf R$ be a one-parameter family of linear operators on $\cal H$. Strong continuity for $U(t)$ is defined as the condition that $$ \lim_{t\to ...
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4answers
57 views

norm in $L^2$ and $l^2$

let $f:R \rightarrow C $ , $f \in L^2 $ , also let f be continious and defined everywhere ( "normal" function like $f(t)=e^{-t^2/2}e^{i\xi t}$ not like those functions that is made for some counter ...
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14 views

Sobolev space for classic function approximation?

Hi guys could be any convenience in using a sobolev space instead of square integrable space for function approximation? I know that sobolev space are mostly used for PDE, but i was wandering if ...
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1answer
66 views

Intuition for orthogonality in infinite dimensions

I'm trying to explain orthogonality in inner product function spaces (e.g. Hilbert spaces) intuitively. As main expample, take the $L^2$ inner product given by $$<f,g>_{L^2(I)}:=\int_I ...
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20 views

Weakly convergent subsequence under continuous operator

Suppose we have two Hilbert spaces $H_1,~ H_2$, a linear continuous operator $T:H_1 \to H_2$ and a weakly convergent sequence $u_k\rightharpoonup u$ in $H_1$. Is $Tu_k \rightharpoonup Tu$ in $H_2$ ...
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25 views

difference of projections is a positive operator

Let $p,q:H\to H$ bounded, linear operator on a Hilbert space $H$, such that $p^*=p=p^2$ and $q^*=q=q^2$ ($p^*$ is the adjoint of $p$, for q the same. YOu call $p$ and $q$ a projection). Let ...
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26 views

The inner product spaces and linearity in probability

Consider a class of inner product spaces $$\langle \cdot,\cdot\rangle_{{\lambda}\in \Lambda}: R^n\times R^n\to R$$ parameterized by $\lambda \in \Lambda=\Delta(\{w_1,....,w_n\})$, the set of all ...
2
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47 views

Argument for extenstion of the Fourier transform

Could anyone please point out if there is any mistakes in the following arguments for the extension of the Fourier transform from $\mathcal{S}(\mathbb{R})$ to $L^2(\mathbb{R})$: "Since the compactly ...
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25 views

Extension of the Fourier transform, proof-read

I'm writing a Bachelor-thesis in mathematics which is to be submitted in a couple of days, and would be thankfull if the following arguments concerning the extension of the Fourier transform from the ...