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

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Injectivity of bounded operator on Hilbert space

Let $T$ be a bounded linear operator on a Hilbert space $H$. It is cleat that if there exists $\delta>0$, such that $$ \langle T^{*}Tx,x\rangle\ge \delta ||x||^2 $$ for all $x$, then $T$ is ...
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A variant of Bessel's ineaquality, without orthonormality

Let $\{x_n\}$ be a fixed sequence in a Hilbert space $H$, such that there exists a bounded linear map $T$ from $\mathcal{l}^2$ to $H$ sending $(a_n)$ to $\Sigma_n a_n x_n$. I've been unsuccesfully ...
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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 ...
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A characterization of Bessel sequences in a Hilbert space

I've shown that if for a sequence $\{f_{n}\}_{n=1}^{\infty}$ in a Hilbert space $H$ we have $$\sum_{n=1}^{\infty}|\langle f,f_n\rangle|^{2}< \infty$$ for all $f\in H$ (i.e., it is a Bessel ...
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Can one define a functional on a Hilbert space based on its action on a Hilbert basis?

I know that the actions of a functional on a vector space can be uniquely described by the value the functional takes on each element of a (Hamel) basis. My question is, in a Hilbert space, would ...
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Weak Convergence implies boundedness and componentwise convergence

Let $\ell^2$ be the set of real number sequences $\{a_n\}$ such that $\sum a_n^2 <\infty$. Let $\langle a_n,y\rangle \rightarrow \langle a,y\rangle$ for some $a\in \ell^2$ and for all $y\in ...
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a Representation of C*-algebra

I have a quick question about a representation of c*-algebra. So a representation of c*-algebra $A$ is a *-homomorphism $\pi : A \rightarrow B(H)$ where $B(H)$ is a set of bounded operators on some ...
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158 views

One of these two operators is not invertible

I have a Hilbert space $H$ and a bounded self-adjoint operator $T$ on it with $||T||=1$. I've been trying to show that at least one of $I+T, I-T$ are not invertible, but I haven't been able to make ...
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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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Sufficient conditions on integration kernel for continuity of the integral operator

Suppose that we have a measure $d\mu(v)=e^{-|v|^2}dv$ on $\Bbb R^d$. We define a linear operator $$T[f](u)=\int_{\Bbb R^d} |u-v|^\beta d\mu(v).$$ I want to establish conditons on $\beta\in\Bbb R$ so ...
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Unitary G-module

I'm not sure if I understand this sentence correctly: "By a unitary G-module we will mean a Hilbert space W on which G acts by means of a strongly continuous unitary representation". $G$ is a ...
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1answer
16 views

Normal Operators: Construction

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: $$A:=N(1+N^*N)^{-1}\in\mathcal{B}(\mathcal{H})$$ ...
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59 views

Linear algebra references for a deeper understanding of quantum mechanics

I'm a graduate student studying quantum mechanics/quantum information and would like to consolidate my understanding of linear algebra. What are some good math books for that purpose?
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42 views

Orthogonal in inner product space

Let $(X,<.>)$ is an inner product space prove that $x$ and $y$ are orthogonal if and only if $||x+αy|| \ge ||x||$ for any scalar $α$ . The first direction if $x$ and $y$ are orthogonal ...
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Modulus: Invariant Domain

Problem Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider an operator: $$M:\mathcal{H}\to\mathcal{K}:\quad \|M\|=1$$ Regard dense subspaces: ...
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Show trigonometric function are complete on $L^2[0,2\pi]$

The proof is in the book but I couldn't understand it. Will appreciate your help. My doubts are in blue. Proof: Suppose $f(\theta)$ is any continuous, $2\pi$ periodic function ...
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1answer
19 views

Projection theorem for nonclosed subspaces

Is there a substitute for the projection theorem for Hilbertspaces (if $M$ is a closed subspace of $H$ then $H = M \oplus M^\perp$) in the case that $M$ is a linear subspace of $H$ which is not ...
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Banach space it isn't Hilbert space [duplicate]

How can give me two or three examples about Banach spaces which it is not Hilbert spaces with proof ( I mean why it isn't Hilbert spaces ) ?
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31 views

Can someone explain this problem I am having with the proof of the Riesz-Fischer theorem

Here is the form of the theorem I have; Let $\{e_n\}_{n=1}^{\infty} \in H$ be an orthonormal set (H a Hilbert space with inner product $(.,.)$) and let $(a_n)_{n=1}^{\infty}$ be an arbitrary sequence ...
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$X$ is inner product space then its completion is Hilbert space?

I have trouble finding a way to prove that the completion of my innerproduct space $X$ is a Hilbert space. How do I know that the norm on the completion of $X$ is induced by an innerproduct? Thanks ...
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Bounds for spectrum of self-adjoint operator on Hilbert space

$A$ is an self-adjoint bounded operator on Hilbert Space $H$, that is for all $x,y\in H$, $(Ax,y)=(x,Ay)$. $(~,~)$ is inner product of H. $$ m=\inf\limits_{||x||=1}(Ax,x) ~~~~~ ...
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Selfadjoint Operators: Relative Boundedness

Problem Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard an operator: ...
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Selfadjoint Operators: Empty Spectrum

Can a selfadjoint operator have empty spectrum? (As far as I remember, yes; but just to be sure.) The point is that if so then the closure of its spectrum cannot equal the convex hull of its ...
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Subspace of $L^p(X,\Sigma,\lambda)$

Consider $R$-valued functions in $L^p(X,\Sigma,\lambda)$, where $X=X^1\times X^2$, $\Sigma=\Sigma^1\times \Sigma^2$ and $\lambda=\lambda^1\times \lambda^2$ For given $i$, does the subsapce $M=\{f\in ...
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Møller Operators: Functional Calculus

Problem Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider Hamiltonians: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ $$K:\mathcal{D}(K)\to\mathcal{K}:\quad K=K^*$$ and a bounded ...
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288 views

Double orthogonal complement of any closed subspace is it self

Let $H$ be a pre-Hilbert space such that any closed sub space $M \subset H$ has the property $M^{\bot \bot}=M$. Prove that $H$ is a Hilbert space (ie, prove that $H$ is complete) My attempt: As ...
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185 views

Proving that a sequence in $L^2(\mathbb R)$ is relatively compact

I have a bounded sequence $\{f_n\}_n$ in $L^2(\mathbb R)$ such that $\mbox{supp } f_n$ is uniformly bounded and $$ \int_{\mathbb R} x^2 |\Theta_n(x) (F f_n)(x)|^2 dx \leq C^2 $$ for all $n$, where ...
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29 views

Help needed in determining if statement is true

Let $H$ be a Hilbert space, and $(x_n)_{n=1}^\infty$ be a sequence in $H$ with $x_n\rightharpoonup x\in H$ weakly. Then $\|x_n\|\to\|x\|$. I found this in one of my textbook; my question is if ...
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Fourier coefficients in a non-separable Hilbert space

Let $H$ be a non-separable Hilbert space. Let $\{ \phi _\alpha\} _{\alpha \in A} $ be a orthonormal system on $H$. Show that for every $x\in H$, there are only countably many Fourier coefficients, ...
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Need help for construct DLSQ-spline in $B$-form

In Schumaker, Larry L. "Computing bivariate splines in scattered data fitting and the finite-element method." Numerical Algorithms 48.1-3 (2008): 237-260 (Link to journal, link to author page for ...
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34 views

When is the Laplace Beltrami Operator self-adjoint?

The Laplace-Beltrami operator is an operator which is the typical example of a self-adjoint operator in $L^{2}$. I am wondering if this is also true for other Hilbert spaces $W^{k,2}$. If this is ...
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distances in $l^2$

This problem was posed by my friend and he said I may want to use some combinatorial set theory: Can you give me example of an uncountable $X \subseteq l^2$ (the Banach space of square summable ...
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Problem 8 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein and Shakarchi's Real Analysis

The following is problem 8 from chapter 4 ("Hilbert Spaces: An Introduction") of Stein and Shakarchi's Real Analysis. Suppose $\{t_k\}$ is a collection of bounded operators on a Hilbert space $H$. ...
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Hilbert-Schmidt and compact operators

I am new to this site and i dont really know how to ask questions properly, so i am really sorry if i did something wrong. My question is if there is a way to prove that a Hilbert-Schmidt operator is ...
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Identity Operator can be uniformly approximated by orthonormal basis

Let $H$ be a separable Hilbert space with orthonormal basis $e_1, e_2, ...$. I know that for any $x \in H$, we have $$\|x\|^2 = \sum\limits_n \|\langle x, e_n \rangle\|^2$$ and in fact $x = ...
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RKHS vs RKHS from sobolev spaces.

Which is more desirable in terms of solving a differential equation? Constructing an RKHS from sobolev space (essential reproducing kernels are of infinite support). Or directly choosing ...
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1answer
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Characterization of positive elements of a sub-C*-algebra of $B(H)$

Let $A$ be a non-unital sub-C*-algebra of $B(H)$. I want to show that if $T\in A$ and $\lambda \in \mathbb{C}$ are such that $T+\lambda I_H\geq 0$ then $T$ is self-adjoint and $\lambda \geq 0$. Let ...
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23 views

Partial Isometries: Subspaces

Note: This thread is not to gain reputation!!! Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider a bounded operator: $$W:\mathcal{H}\to\mathcal{K}:\quad\|W\|<\infty$$ Then a partial ...
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States: Density

Problem Given a Hilbert space $\mathcal{H}$. Regard the CAR-algebra: $$\{a(\eta),a(\zeta)\}=0\quad\{a(\eta),a(\zeta)^*\}=\langle\eta,\zeta\rangle$$ Consider a density: ...
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How is the Point Spectrum of a Compact Operator Countable?

I'm working on understanding a proof that if an operator $A$ on a Hilbert space $\mathcal{H}$ is compact, then show that $\sigma(A) - \{0\} \subseteq \sigma_p(A)$. If you're not familiar with this ...
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If $H$ is a Hilbert space, does the coordinate projection $\pi :H\oplus H\rightarrow H$ take closed subspaces to closed subspaces?

Here $H\oplus H$ has the product topology, which is induced from the "$\ell^2$-norm" $\|(x,y)\|:=\sqrt{\|x\|^2+\|y\|^2}$. This is indeed a Hilbert space via the inner product $\langle (x,y), ...
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Spectrum of simple multiplication operator on $L^2(0,1)$

I'm trying to calculate the spectrum of the linear operator $T: L^2(0,1) \to L^2(0,1)$ given by $T(f) \to tf(t)$. I've found a few facts about this operator but I'm still struggling to find the exact ...
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What Do Hilbert Spaces Look Like?

For any vector space $V$ over $\mathbb{C}$, let $X$ be a set whose cardinality is the dimension of $V$. Then $V \cong \bigoplus\limits_{i \in X} \mathbb{C}$ as vector spaces. Is there a similar ...
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The cardinality of dense subsets of infinite-dimensional Hilbert spaces

If $H$ is an infinite-dimentional Hilbert space, then does $\dim H$ coincide with the smallest cardinal of a dense subset of $H$?
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question of hilbert spaces

let A be a bounded linear functional on the subspace M of the Hilbert space H , show that there exists a unique extension of A to H having the same norm ?
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About the largest eigenvalue of Hilbert-Schmidt integral operators

Let $\Omega$ be an open set of $\mathbb{R}^d$ and $K \in L^2(\Omega\times \Omega)$ such that for almost all $x,y \in \Omega$ : $K(x,y)=K(y,x)$ $K(x,y)>0$ One can show that under these ...
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Are continuous bounded functions a subspace of $L^2$?

I have a problem where I need to work with functions that are square-integrable, bounded and continuous, i.e. the space $ L^2 \supset X = \left\{ f \in L^2 \mid f \text{ bounded, continuous}\right\} ...
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45 views

Projection on closed subspace of $L^p$, $1<p<\infty$

Let $1<p<\infty$ and $K$ be a closed subspace of $L^p(X, \mathcal{M}, \mu)$. If $f\in L^p$ then there exists a unique $h\in K$ such that $||f-h||_p$ equals $$ \text{dist}(f,K)=\inf_{g\in ...
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26 views

Notation in Reed/Simon Vol. IV (and possibly an earlier volume)

I'm wondering if there are any mathematical physicists/analysts out there that can help me with some notation I've seen in Reed and Simon's books on analysis. Unfortunately I don't have time to read ...
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Projection on closed subspace of $L^1$, $L^{\infty}$

For $p=1,\infty$ let $K$ be a closed subspace of $L^p(\mathbb{R},m)$. According to this question, it should be easy to find examples of $K$ and $f\in L^p(\mathbb{R},m)$ such that there exists a ...