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

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1answer
19 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
21 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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38 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
40 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
10 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 ...
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1answer
28 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
21 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 ...
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1answer
16 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 ...
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1answer
27 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} ...
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2answers
36 views

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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2answers
173 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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1answer
33 views

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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0answers
5 views

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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1answer
37 views

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
25 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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1answer
72 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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1answer
43 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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22 views

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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2answers
54 views

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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0answers
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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1answer
27 views

$X$ 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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10 views

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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16 views

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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2answers
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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0answers
20 views

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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31 views

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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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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30 views

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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36 views

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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40 views

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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1answer
28 views

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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14 views

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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15 views

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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29 views

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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2answers
35 views

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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1answer
27 views

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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1answer
24 views

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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3answers
93 views

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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1answer
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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30 views

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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21 views

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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9 views

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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31 views

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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1answer
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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1answer
19 views

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 ...
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1answer
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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1answer
34 views

Is this a Hilbert space?

For $n\geq 2$, we let $\mathcal{H}$ be the complex vector space of all complex-valued functions on $[0,1]$ such that (a) $f(0)=0$, (b) for $1\leq k\leq n-1$, $f^{(k)}$ exists everywhere and is ...
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0answers
46 views

help in proving equivelant statements of $f=\sum\limits_{i=1}^\infty \langle f,\phi_i \rangle \phi_i \space \forall f\in H$

Let $H$ is a hilbert space. $\{\phi_i\}_{i=1}^\infty$ is an orthonormal set A set $\{\phi_i\}_{i=1}^\infty$ is complete in $H$ if any of the following statements hold: $f=\sum\limits_{i=1}^\infty ...
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3answers
73 views

Infinite dimensional spaces other than functional spaces

"Functional analysis" is the study of infinite dimensional spaces equipped with inner product, norm, topology...etc. The most interesting spaces are the spaces of functions/operators and sequences. I ...