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

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1answer
32 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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0answers
26 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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0answers
17 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: ...
3
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0answers
56 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 ...
3
votes
2answers
37 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), ...
0
votes
1answer
66 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 ...
2
votes
1answer
31 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
105 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 ...
4
votes
1answer
191 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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0answers
22 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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21 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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0answers
34 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\} ...
1
vote
1answer
44 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 ...
2
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1answer
24 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 ...
2
votes
1answer
54 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 ...
2
votes
1answer
37 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
49 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 ...
2
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3answers
81 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 ...
5
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1answer
191 views

Proof involving strongly continuous semigroups.

Let $ (T(t))_{t \geq 0} $ be a $ C_{0} $-semigroup on a Hilbert space $ X $ with an infinitesimal generator $ A $, and let $ \rho \in (0,1) $. I want to prove that $ \displaystyle \sup_{t \geq 0} \| ...
1
vote
2answers
115 views

Compact Operators: Trace

Given a Hilbert space $\mathcal{H}$. Consider a bounded operator: $$A:\mathcal{H}\to\mathcal{H}:\quad\|A\|<\infty$$ Regard ONB's: ...
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1answer
35 views

Is a separable Hilbert space spanned by a countably dense subset?

Hi I am reading a proof in my functional analysis notes and there is a step I don't really understand; Since H (infinite dimensional Hilbert space) is separable it contains a countable sense subset ...
0
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1answer
19 views

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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1answer
22 views

Prove that a) $(span(M))^\bot=M^\bot$ b) $(\overline{M})^\bot=M^\bot$

Let $X$ be an inner product space.$M\subset X$. Prove that a) $(span(M))^\bot=M^\bot$ b) $(\overline{M})^\bot=M^\bot$ My Work and problems: a) Clearly $(span(M))^\bot\subset M^\bot$. Now let ...
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vote
1answer
105 views

If $u\in H^2(R^n)$, how to prove $\|D^2u\|_{L^2}$ is equal to $\|\Delta u\|_{L^2}$ using Fourier transforms?

Problem: If $u\in H^2(R^n)$, how to prove $\|D^2u\|_{L^2}$ is equal to $\|\Delta u\|_{L^2}$ using Fourier transforms? My first question: Is it right to prove this using integration by parts as ...
0
votes
1answer
55 views

$x\perp y$ iff $\|x+\lambda y\|\geq \|x\|$ for all scalars $\lambda$

Show that in an inner product space $X$ a) $x\perp y$ iff $\|x+\lambda y\|=\|x-\lambda y\|$ for all scalars $\lambda$ b) $x\perp y$ iff $\|x+\lambda y\|\geq \|x\|$ for all scalars $\lambda$ My ...
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0answers
14 views

Orthogonality among elements of $\mathcal{H}_2$ and $\mathcal{H}_2^{\perp}$

Let $\mathcal{L}(j\mathbb{R})$ a Hilbert space of matrix-valued functions on $j\mathbb{R}$ which consists of all complex matrix functions $F$ such that the following integral is bounded ...
2
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1answer
45 views

Dual space of a closed subspace of a Hilbert space

I'm reading Girault and Raviart's book concerning Finite Element Methods for Navier-Stokes equations, and they use in the proof of one result, the following argument: As $V=\{v\in H_0^1(\Omega)^N; ...
0
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1answer
32 views

Quadratic Functional Differentiability

I would like to solve the following: Let $T$ be a self-adjoint bounded operator on a Hilbert space $H$. Consider the quadratic functional $\Phi$ defined by: \begin{equation} ...
0
votes
1answer
26 views

$L^2$ integrability

Consider the function $f(x)=x^{-1/2}$ for $0<x<1$, and $0$ else. Let $\{r_n\}_{n=1}^{\infty}$ an enumeration of the rationals. Let $F(x)= \sum_{n=1}^{\infty}2^{-n}f(x-r_n)$. Prove that $F(x)$ is ...
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1answer
27 views

Are the operators $X$ and $Y$ equal to one another?

Suppose we've got two linear maps $\ X:\mathcal{H}\rightarrow\mathcal{H}\ $ and $\ Y:\mathcal{H}\rightarrow\mathcal{H}$, where $\mathcal{H}$ is some finite-dimensional Hilbert space. Let's say ...
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0answers
62 views

Partial Trace of Density Operator

Before stating my question I present my motivation: to learn more about the tensor product. Now, quantum mechanics assigns a Hilbert space to each physical system as a postulate of the theory. ...
0
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1answer
29 views

Bijective bounded linear operator is invertible

The following is an exercise from Halmos book "A Hilbert space problem book" : Exercise: If $H$ and $K$ are Hilbert spaces, and if $A$ is a bounded linear transformation that maps $H$ one to one and ...
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2answers
41 views

$\int_{\mathbb R}|f(x)|^{2} dx <\infty \implies \sum_{m\in \mathbb Z}\int_{m-\beta}^{m+\beta}|f(x)|^{2} dx <\infty$?

Let $f\in L^{2}(\mathbb R),$ that is, $\int_{\mathbb R}|f(x)|^{2} dx <\infty,$ and $\beta>0.$ My Question: Is it true that that: $\sum_{m\in \mathbb Z}\int_{m-\beta}^{m+\beta}|f(x)|^{2} dx ...
0
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0answers
31 views

Differential of a mapping $F:X\rightarrow X$

I'm reading a paper and says: The differential of a mapping $F:X\rightarrow X$ with respect to $v$ is: $D_vF(x)\in{\cal L}(X)$ where $H$ is a Hilbert space and ${\cal L}(X)$ stands for ...
0
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1answer
26 views

Z-transform and$ H_2$ space

The following is from the preliminaries of a paper. Let $\mathbb{D} = \{z \in \mathbb{C} : |z|<1\}$ be the unit disc of complex numbers. A function $G:(\mathbb{C} \cup \{\infty\})\backslash ...
3
votes
1answer
76 views

Dense subset of Hilbert space has trivial orthogonal complement

If $D$ is a dense subset (not subspace) of a Hilbert space then is the orthogonal complement of $D$ equal to $\{ 0 \}$? This is true if $D$ is a subspace but if you only know that $D$ is a subset ...
2
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0answers
38 views

A sequence of strongly continuous one-parameter unitary groups

Suppose that for a sequence $\{A_n\}_n$ of bounded self-adjoint operators in a Hilbert space $\mathcal H$ we have $e^{itA_n} \to e^{itA}$ strongly, for all $t \in \mathbb R$, where $A$ is a (possibly ...
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vote
1answer
51 views

Convergence of a series

Can someone help me how to do the following? Let $H$ be a Hilbert space with inner product $\langle \cdot,\cdot \rangle$, and orthonormal basis $(e_n)$. For $x,y \in H$ prove that the series in the ...
4
votes
1answer
39 views

Show that in a complex Hilbert space, T normal bounded linear operator, $\| T^2 \| =\| T \| ^2$

So, as a part of a problem, I've been asked to prove that if $H$ is a complex Hilbert space and $T\in L(H)$ is normal, then $\| T^2 \| =\| T \| ^2$ (Operator norm) Context: This is part (b) in a ...
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votes
1answer
57 views

Resolvent: Decay Behavior

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote resolvent set: ...
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0answers
49 views

Every Hilbert space is connected

Let $H$ be a Hilbert space. Proving $H$ is connected, suppose $\{e_i\}_{i\in I}$ is a orthogonal basis of $H$. Thus $H=\bigoplus_{i\in I} \Bbb C e_i$. Clearly $\Bbb Ce_i$ is connected for every $i$, ...
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1answer
31 views

Bijection between a Hilbert and a Banach space

I am working on a question where I have to show that for a Hilbert space $\mathscr{H}$, and closed linear subspace $Y \subset \mathscr{H}$, $\mathscr{H}/Y$ is a Hilbert space, isomorphic to ...
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1answer
53 views

Eigenvectors of operators on a tensor product Hilbert Space

Suppose I have finite dimensional Hilbert spaces $V$, $W$, and an operator $A$ acting on vectors in $V$ such that it has eigenvectors/values $Ax_a=\lambda_ax_a$. In the tensor product space I want to ...
2
votes
1answer
62 views

Spectral Measures: Core Lemma

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a dense domain: ...
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votes
1answer
25 views

Question about ortogonality on $L^2(\Omega)$

Let $u\in L^2(\Omega)$. Is the following proposition true? $\big(\forall v\in H^1_0(\Omega)\big)\quad (u,v)_{0,\Omega}:=\displaystyle\int_\Omega uv=0$ then $u=0$ ? where $H_0^1(\Omega)$ are the ...
0
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1answer
12 views

Characterization of product of dual dual Hilbert spaces.

Let $X_i$ be Hilbert spaces and $X_i'$ its dual spaces, with $i=1,2$. Let $F\in(X_1\times X_2)'$. Prove that exists $F_1\in X_1'$ and $F_2\in X_2'$ such that ...
0
votes
2answers
109 views

Spectral Measures: Scale Embeddings

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}N\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
0
votes
1answer
100 views

Convergence of the spectrum under norm resolvent convergence

Suppose $\{A_n\}$ is a sequence of self-adjoint operators in a Hilbert space $\mathcal H$, and $A$ is a self-adjoint operator, with $A_n \to A$ in norm resolvent sense. Since $A_n \to A$ in strong ...
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vote
1answer
54 views

Relation between residual spectrum and point spectrum.

Suppose T is a bounded operator on a Hilbert space. Show that if λ is in the residual spectrum of T, then $\bar{λ}$ is in the point spectrum of the adjoint. Here is what I think needs to be done. ...
3
votes
1answer
16 views

A sequence $L_n$ of compact bounded linear transformations on a hilbert space defines a convergent subsequence in each $L_n$ for a bounded sequence?

Let $L_n:\mathcal{H}\to\mathcal{H}$ be a sequence of compact bounded linear transformation on a Hilbert space $\mathcal{H}$, and $h_m$ be a sequence in $\mathcal{H}$. Since each $L_n$ is compact, ...