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

learn more… | top users | synonyms

0
votes
0answers
9 views

Generalized Mixed Models and Hilbert Spaces

I recently came across the derivation of the normal equations for linear regression using Hilbert Spaces and projection theorem, and thought it was pretty cool. After doing a lot of googling, it seems ...
4
votes
2answers
64 views

Are those two definitions of orthogonal projection equivalent in a general Hilbert space?

I am taking a graduate level course in probability and we started off with some results in functional analysis. One thing that I feel I do not understand properly is the definition of an orthogonal ...
0
votes
0answers
14 views

$A$ and $A^*$ dissipative implies $D(A) \subset H$ is compact embedding

For selfstudy purpose I want to show the following: $H$ Hilbertspace, $D(A)$ dense subspace of $H$, $A\colon H \supset D(A) \to H$ linear closed dense defined operator. If $A$ and $A^*$ are both ...
0
votes
2answers
37 views

Complete orthonormal system in a finite dimension Hilbert space

I have to solve the following problem of functional analysis. Let $H$ be a Hilbert space of dimension $N$. Prove that every complete orthonormal system in $H$ has $N$ elements and that $H$ is ...
1
vote
0answers
71 views

Results about Hilbert-Sobolev space with homogeneous boundary condition.

I am currently reading works about numerical method for solving differential equations. The main setting of the work revolves on the space $H^m_0[0,1]$, which is defined by $$ H^m_0[0,1]:=\{f\in ...
1
vote
0answers
30 views

Exercise on Hilbert spaces and complete orthonormal systems

Let $H$ be a Hilbert space of finite dimension $N$. Prove that every complete orthonormal system in $H$ has $N$ elements and that $H$ is linearly isometric to $\mathbb{R}^N$. I can't start with this ...
0
votes
1answer
15 views

$T+i\operatorname{Id}$ is an isomorphism for self-adjoint $T$

Let $T:H\to H$ be a self-adjoint continuous operator on a complex Hilbert space. Claim: $T+i\operatorname{Id}$ is an isomorphism and $\|(T+i\operatorname{Id})^{-1}\|\leq 1$. A few observations: ...
2
votes
1answer
29 views

$\operatorname{span}\{x_n:n\in \Bbb N\}$is dense if $\sum_{n=1}^\infty \|x_n-e_n\|^2<1$

Let $H$ be a Hilbert space with orthonormal basis $\{e_1,e_2,\cdots\}$. Suppose $(x_n)$ is a sequence in $H$ with $\sum_{n=1}^\infty \|x_n-e_n\|^2<1$. Claim: The span of the $x_n$ is dense in ...
1
vote
0answers
22 views

A naïvely constructed extrapolation of a self-adjoint operator. Is it self-adjoint?

Let $\mathcal{H}$ be a real Hilbert space and let $A\colon D(A)\subset \mathcal{H}\to \mathcal{H}$ be an unbounded operator. Consider also a Hilbert triple $$ \mathcal{H}_+\subset \mathcal{H}\subset ...
1
vote
1answer
53 views

Is there a relation between Cartesian and tensor product of function spaces and function factorizability

H1 and H2 are two Hilbert spaces represented by a function space, say f1(x1) and f2(x2) are its vectors. If H3 is tensor product of H1 and H2 I assume one can say that f(x1,x2) now represents vectors ...
1
vote
2answers
23 views

If the sum of two weakly convergent sequences strongly converges, do the summands strongly converge?

Let $a_n \rightharpoonup a$ and $b_n \rightharpoonup b$ weakly in $H$, a Hilbert space, and suppose that $a_n + b_n \to a+b$ strongly in $H$. Is it true that $a_n \to a$ and $b_n \to b$ strongly? I ...
2
votes
2answers
50 views

What's the value of $\alpha$ satisfying $||f'||^2\ge \alpha||f||^2$? [duplicate]

I am reading a paper about numerical analysis of a certain method for solving operator equation. Let our Hilbert space be $L^2[0,1]$, we define the subspace $D\in L^2[0,1]$ by $$ D:=\{f\in ...
1
vote
1answer
32 views

Continuity of inner product and change of limit order

First of all, please note that the specific context of ergodic theory could possibly not matter and this could reduce to a simply a question about Hilbert Space. As a part of a proof i'm working on, ...
0
votes
0answers
34 views

Square integrable functions on the unit ball

In one dimension, the space $L^{2}([0,1], dx)$ of complex-valued square integrable functions on $[0,1]$ is well known to be separable, and sine and cosine provide an explicit Hilbert basis for it. My ...
7
votes
1answer
256 views

Holomorphy of a function with values in a Hilbert space

Denote by $\mathbb C^\infty $ the Hilbert space $\ell^2(\mathbb C)$. Fix $1\leq N,M \leq \infty$, and let $U$ be an open subset of $\mathbb C^N $. Following Mujica's book "complex analysis in Banach ...
1
vote
2answers
29 views

Exercise on separable Hilbert spaces and orthonormal system

Let $H$ be a separable Hilbert space and $\{e_n\}$ a complete orthonormal system of $H$. Prove that, if $\{y_k\}$ is a bounded sequence in $H$, the condition $\lim_{k\rightarrow\infty}(e_n, y_k)=0$ ...
0
votes
1answer
18 views

Prove that conditions are equivalent

$ X $ is unitary space, $ x,y \in X $. Prove that following conditions are equivalent: $ x \perp y $ $ ||x|| \leq ||x+ty|| $ $ t \in C $ $ ||x+ty||=||x-ty|| $ $ t \in C $ Unfortunatelly, I'm ...
1
vote
1answer
24 views

Bergman space norm in terms of coefficients

I am interested in the Bergman space $A^2$ on the unit disc. According to the Wikipedia article on Bergman spaces, if we have $f(z)= \sum_{n=0}^\infty a_n z^n \in A^2$ then $$\|f\|^2_{A^2} := ...
2
votes
0answers
25 views

Compactness in Hilbert spaces

Let $H$ be a Hilbert space with orthonormal basis $\{h_n:n\in \Bbb N\}$. Let $P_n$ be the orthogonal projection to $\operatorname{span}\{h_1,\cdots, h_n\}$. Claim: A bounded subset $U\subset H$ is ...
0
votes
2answers
28 views

Exponent of an Exponential Operator

There is a problem in my textbook that asks me to prove the following: For a bounded operator $A$ on a Hilbert space, prove that: $$(e^A)^n = e^{An} $$ for any natural number, $n$. However upon ...
4
votes
0answers
49 views

Submultiplicative Hilbert space norm on $B(H)$

Let $H$ be a complex Hilbert space and let $B(H)$ denote the space of bounded linear operators $H \to H$ equipped with operator norm: $$ \lVert T \rVert = \sup\big\{ \lVert Tx \rVert \: : \: \lVert x ...
0
votes
0answers
20 views

Show $L$ is a closed linear subspace of $H$

Let $H$ be a Hilbert space and linear operator $P:H \rightarrow H$ satisfy $<Px,y>=<x,Py>$ for all $x,y \in H$ and $P^2=P$. We can use the fact that $Px \perp (x-Px)$ for every $x \in H$ ...
1
vote
2answers
36 views

Invariant subspaces for this linear extension of operators

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 $T(e_k)=e_{k+1}$ , $(k=1,2,\cdots)$ and then linearly and ...
1
vote
3answers
50 views

Show $Px \perp (x-Px)$ and prove $||P|| \leq 1$

Let $H$ be a Hilbert space and linear operator $P:H \rightarrow H$ satisfy $\langle Px,y\rangle = \langle x,Py\rangle$ for all $x,y \in H$ and $P^2=P$. (a) Show that $Px \perp (x-Px)$ for every $x ...
3
votes
1answer
89 views

Proving an isometric dilation of a non unitary operator on Hilbert space implies infinite dimensional space involving matrices

I have been given this exercise in my Operator theory class dealing with operators on Hilbert spaces, which reads as follows: Let H be a Hilbert space. We are to prove, in two distinct ways, that ...
1
vote
1answer
18 views

Hilbert space, functional analysis

Let $X$ and $Y$ be closed subspaces of a Hilbert space $H$. Assume that dim $X < \infty$, and dim $X$ < dim $Y$. Show that $X^{\perp} \cap Y \neq \{0\}$. I want to proof it by contradiction. ...
2
votes
2answers
63 views

Spectral Measures: Integrability

I really need this as tool for other threads! Given a Hilbert space $\mathcal{H}$. Also a Borel space $\Omega$. Consider a spectral measure: $$E:\mathcal{B}(\Omega)\to\mathcal{P}(\mathcal{H}):\quad ...
1
vote
1answer
39 views

Eigenvectors Operators and Unilateral Shifts

We showed that a (non-zero) compact self-adjoint operator on a Hilbert space always has an eigenvector. Let $V:l^2(\mathbb{N})\to l^2(\mathbb{N})$ be the unilateral shift, the unique bounded operator ...
0
votes
3answers
55 views

Vectors in a Hilbert space are countably supported with respect to any orthonormal basis

Let $\{e_i\}_{i\in I} \subset \mathcal{H}$ be an orthonormal set in the Hilbert space $\mathcal{H}$. For any vector $x\in \mathcal{H},$ let $$I_x=\{i\in I|\,\langle x,e_i\rangle \neq 0\}.$$ How can we ...
5
votes
0answers
34 views

Bounded Operators: Topological Dual

Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider the bounded operators: $$\mathcal{B}(\mathcal{H},\mathcal{K}):=\{T:\mathcal{H}\to\mathcal{K}:\|T\|<\infty\}$$ Regard the linear ...
1
vote
0answers
22 views

Approximating step functions by Haar wavelets

Let $\psi = \chi_{[0,1/2)} - \chi_{[1/2,1)}$, then $\psi_{n,k}(t) = 2^{n/2}\psi(2^nt-k)$ with $n \in \mathbb{N}$ and $k \in \{0,1,\dots,2^n-1\}$ defines the Haar-Wavelets on $L^2(0,1)$. Let $S$ be the ...
1
vote
1answer
21 views

Inner Product on Sobolev Space with p=2

Wikipedia defines the Sobolev Space: $H^{s,p}(\mathbb{R}^n)= \left\{f \in L^p(\mathbb{R}^n): \mathcal{F}^{-1}[(1+|k|^2)^{\frac{s}{2}} \mathcal{F}f] \in L^p(\mathbb{R}^n) \right\}$ Where $s \in ...
2
votes
2answers
47 views

Question on operator theory classes of operators on Hilbert spaces

I was recently tackled by this in my class on operator theory dealing with operators on Hilbert spaces: We are to find and prove the inclusion relations between the classes of operators: ...
0
votes
1answer
31 views

Norm estimation of a function on Hilbert space implies linear transformation has closed range

I want to prove the following fact: Let $T: X\to Y$ be a bounded linear transformation between two Hilbert spaces $X$ and $Y$. Show that if there exists a constant $C$ such that $\|f\| \leq C ...
2
votes
1answer
34 views

Exponential of a self-adjoint operator

Let $\mathcal{H}$ be an Hilbert space. Firstly, I shall define some notions as their definitions may vary: A spectral resolution is a function $E:\mathbb{R}\to\mathcal{L}(\mathcal{H})$ (the space ...
1
vote
0answers
40 views

orthonormal basis in $L^2$ space

Let $\{\phi_i (x)\}_{i=1}^\infty$ be an orthonormal basis for $L^2 (S)$. Prove that $\{\psi_{ij} (x,y) = \phi_i (x) \phi_j (y)\}_{i,j=1}^\infty $ is an orthonormal basis for $L^2 (S \times S)$. Thanks ...
0
votes
0answers
20 views

Are quantum operators associative?

Let H be the Hamiltonian representing the total energy of the potential and kinetic component. But because all classical dynamical variables can be written as a function of position, x, and momentum, ...
2
votes
1answer
71 views

Sobolev spaces of sections of vector bundles

Suppose that $X$ is a compact (smooth) $n$-manifold and $E \to X$ be a rank $N$ smooth (complex) vector bundle. Choose finite covering $(U_i)_i$ by domains of the charts $\varphi:U_i \to V_i \subset ...
6
votes
1answer
41 views

Completeness of derivatives of Hilbert basis with respect to a parameter

Let us take a Hilbert basis $\left|x_\lambda\right >$ in a Hilbert space $\mathcal{H}$, i.e. the $\left|x_\lambda\right >$ are a complete, orthonormal set of vectors. The subscript indicates ...
2
votes
1answer
34 views

Characteristic functions of infinite dimensional random elements

I am trying to understand if it is possible to prove the convergence in distribution of a sequence of infinite dimensional random elements using characteristic functions. Suppose that ...
5
votes
1answer
132 views

Explicit characterization of dual of $H^1$

Let's start by some well-known facts: $H^1(\mathbb{R})$ is a Hilbert space, hence there holds the Riesz representation theorem, stating that any linear functional on it can be represented as $L = ...
1
vote
1answer
43 views

Property of bounded linear transformation between Hilbert spaces

I've asked a question on related question in a previous thread, but I wanted to ask a follow up question. If a bounded linear transformation $T: X \to Y$ where $X$ and $Y$ are Hilbert spaces has ...
1
vote
1answer
27 views

Completeness, spanning and orthonormal bases

I am having some difficulty in understanding some concepts regarding Hilbert spaces. I am learning wavelet theory (with regards to signal processing) and am reading up some basics on signal ...
2
votes
2answers
77 views

Proving a variant of closed range theorem on Hilbert space

I've been working on closed range theorem. There are a lot of materials on general Banach spaces, but not much on Hilbert spaces, so I was wondering if I could get some help. I'm trying to prove the ...
3
votes
0answers
38 views

Under what conditions is the resolvent set of a linear operator connected?

Let $(H, \langle \cdot, \cdot \rangle)$ be a Hilbert Space, and assume that $T: H \to H$ is a possibly unbounded linear operator whose domain $D(T)$ is a dense subspace of $H$. As usual, we define ...
0
votes
0answers
24 views

Show that $l^2(\Bbb N , F)$ with equipped norm is not complete.

For $v$ in $l^2(\Bbb N , F)$ with norm on $l^2$ defined as: $\lvert\lvert v\rvert\rvert_{W}= \sum^\infty_{k=1}\frac{\lvert v_{[k]}\rvert}{2^k}$ Show that $l^2(\Bbb N , F)$ with the norm ...
2
votes
1answer
68 views

Every partially defined isometry can be extended to a isometry

I know that the following theorem holds true: Let $S$ be a subset of $\mathbb R^n$, and let $f:S\to \mathbb R^n$ a map such that $d(p,q)=d(f(p),f(q))$ for every $p,q \in S$ (here $d$ is the usual ...
4
votes
1answer
57 views

Understanding the definition of the covariance operator

Let $\mathbb H$ be an arbitrary separable Hilbert space. The covariance operator $C:\mathbb H\to\mathbb H$ between two $\mathbb H$-valued zero mean random elements $X$ and $Y$ with $\operatorname ...
1
vote
0answers
26 views

Does spectral decomposition exist for non-self-adjoint operators?

In theory, if a linear operator $P$ in a Hilbert space $H$ is self-adjoint, we can decompose it as $Pu=\sum_i \lambda_i <\phi_i,u>\phi_i$, where $\phi_i$ is the eigenfunction of $P$. And we can ...
0
votes
0answers
38 views

A question about Hilbert Spaces and convex sets

I am struggling with this and could really do with some help: Let $H$ be a Hilbert space over $\mathbb{R}$, $\{v_n\}$ be a sequence of vectors in $H$, and $C$ be a convex subset of $H$ containing ...