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

learn more… | top users | synonyms

1
vote
1answer
36 views

Compact operator problem: $I-T$ is onto, then $I-T$ is invertible?

I want to show the following: Let $T$ be a compact operator in a Hilbert space $H$. If $I-T$ is onto, then $I-T$ is invertible. Would you show me how to prove this argument? Or please tell me some ...
1
vote
1answer
48 views

Space of $f(0)=f(1)$: Is Hilbert space?

Let $S$ be space consisting of collection of square integrable continuous functions $f:[0,1]\rightarrow\mathbb{R}$ with the constraint $f(0)=f(1)$. So $S$ is an inner product space with the inner ...
0
votes
0answers
25 views

Estimates on the integral of an inner product

Let $X$ be an inner product space. For vector-valued functions $F = (f_1,f_2), G = (g_1,g_2): [0,1] \to X^2$, we define the inner product $$(F, G) = \int_0^1 f_1g_1 + f_2g_2.$$ In particular, $$ ||F||...
0
votes
0answers
4 views

Denseness of polynomial in reproducing kernel Hilbert space

Let $\mathcal H_K$ be a reproducing kernel Hilbert space and $K$ be the associated reproducing kernel on $\mathbb D \times \mathbb D.$ Further assume that $K(z, 0)= 1$ for all $z\in \mathbb D$ and ...
0
votes
0answers
19 views

Basis of convex and concave functions

Let $g(t)$ be a positive function. I'm trying to show that the set $\{ |t|^ne^{\frac{g(t)}{t^n}}\}_{n\in \mathbb{N}}$ is linearly independent in the space of measurable functions on $\mathbb{R}$ (ie: ...
1
vote
0answers
20 views

positive operators

Consider two Hilbert spaces $H$ and $K$ and a bounded operator $T$ on $H\oplus K$. I know that there is a theorem which says the following: $T\ge 0$ (in the operator sense) if and only if $T$ can be ...
1
vote
2answers
35 views

Hilbert space is orthornormality needed for representation?

In a Hilbert space $H$ with countable basis, if I know there is a countable basis $\{h_n\}$ of $H$ then can I express every element $h\in H$ therein as: \begin{equation} h = \sum_n \langle h,h_n\...
1
vote
0answers
27 views

Proving Sobolev space on [0,1] is RKHS

My aim is to prove that the space: $\mathcal{H}$ = {$f:[0,1] \to \mathbb{R}: f\;is\;absolutely\;continuous,\;f(0)=f(1)=0,\;f'\in L^2[0,1]$} is a reproducing kernel hilbert space. Now assuming an ...
4
votes
0answers
35 views

Integration of Hilbert space valued mappings.

TL;DR: Is there a version of the Bochner integral which allows for the integration of isometric embeddings $\phi:X\to H$ from a metric space to a Hilbert space, satisfying $\int_X \|\phi\| d\mu < \...
1
vote
1answer
35 views

Prove that two vectors in Hilbert space are orthogonal

Let $x$ and $y$ be two vectors in a Hilbert space $H$.Prove that $\left\|x+cy\right\|\geq\left\|x\right\|$ for all complex number $c$ if and only if $x$ and $y$ are orthogonal. It's easy to show ...
2
votes
0answers
19 views

continuous and sequentially continuous

If an operator $T: A\rightarrow B$ satisfying for every sequence $\{X_n\}$ weakly converging to $X$, we have $TX_n \rightarrow TX$ in weak topology. Then, is $T$ weak-weak continuous? And in the WOT/...
2
votes
1answer
54 views

For $p(x)\in \Bbb{C}[x]$ such that $\int_{0}^{1}p(x)x^kdx=0$ for $0\le k\le n-1$, show that $p(\lambda)=0\Rightarrow \lambda\in [0,1]$

For a complex polynomial $p(x)\in \Bbb{C}[x]$ of degree $n$ such that $\int_{0}^{1}p(x)x^kdx=0$ for $1\le k\le n-1$, show that $p(\lambda)=0$ means $\lambda\in [0,1]$. I haven't come by any ...
1
vote
2answers
52 views

Prove or disprove: $\{t^{2k}\}_{k=0}^{\infty}$ complete in $L_2[-1,3]$

Is $\{t^{2k}\}_{k=0}^{\infty}$ not complete in $L_2[-1,3]$?(Here, completeness of a system is equivalent to the density of its span) Obviously many polynomials in the domain will be irreleant, but I ...
1
vote
1answer
37 views

What is the Hilbert adjoint operator of this bounded linear operator?

Let $H$ be a Hilbert space, and let $z \in H$. Let $T_z \colon H \to K$, where $K$ is the field of scalars for $H$ and $K$ is either $\mathbb{R}$ or $\mathbb{C}$, be defined by $$ T_z (x) \colon= \...
0
votes
0answers
54 views

Two inner products in one vector space.

Please, can you help me answer the some following questions? In theory functional analysis. At first, I want to consider finitely dimensional vector space V over field K(real or complex). Now, if it ...
0
votes
1answer
21 views

Checking positive definiteness of some matrix

Let $B$ be a bounded self-adjoint operator on the Hilbert space $(\mathcal{H}, \langle \cdot, \cdot \rangle)$ with $0 \not \in \sigma(B)$ and further let $\rho \in \mathbb{R}$ be strictly positive ...
2
votes
2answers
38 views

Weak problem formulation for PDE and boundary conditions

Consider the following example: $$ - \Delta u = f \mbox{ in } \Omega, $$ $$ u = 0 \mbox{ on } \Gamma, $$ Here $\Gamma$ is boundary of $\Omega$. To produce weak formulation we multiply by arbitrary $v$ ...
1
vote
1answer
25 views

If $U$ is a vector subspace of a Hilbert space $H$, then each $x∈H$ acts on $U$ as a bounded linear function $〈x〉$. Is $x↦〈x〉$ injective?

If $H$ is a $\mathbb R$-Hilbert space, then the duality pairing $$\langle\;\cdot\;,\;\cdot\;\rangle_{H,\:H'}:H\times H'\;,\;\;\;(x,\Phi)\mapsto\Phi(x)$$ can be considered as being a mapping $H\times H\...
2
votes
1answer
26 views

Duality Pairing and self-adjoint operators notation

Let $X$ be a Hilbert space and let $X'$ be the dual space of $X$ with respect to the duality pairing $\langle\cdot,\cdot\rangle$. Let $A: X \mapsto X'$ be a bounded linear operator. We assume that $A$...
1
vote
1answer
22 views

Find the spectrum of an operator related to Fourier series

As an exercise, I was told to find the spectrum of the bounded operator $K\in B(L^2[-\pi,\pi])$ defined by $$K\varphi (t)=t\int_{-\pi}^\pi\varphi (x)\cos (x)dx+\cos t\int_{-\pi}^\pi x\varphi(x)dx.$$ ...
3
votes
1answer
18 views

Find eigenvalues and eigenvectors of infinite symmetric matrix of powers of two

Let $a_n=2^{-n}$. What are the eigenvalues and eigenvectors of the $\ell^2$ operator represented by the infinite matrix below? $$A=\begin{pmatrix} a_1 & a_2 & a_3 & \dots \\ a_2 & a_3 &...
1
vote
1answer
16 views

Eigenvalue of every eigenvector is an eigenvalue of element of o.n eigenvector basis

I need to prove (or disprove) that given a bounded operator $A$ on a Hilbert space with orthonormal basis of eigenvectors $\left\{ e_i \right\}$, if $Av=\lambda v$ for $v\neq 0$ then $\lambda $ is an ...
5
votes
1answer
69 views

Functional Analysis by Reed and Simon, chapter 3 exercise 15

Let $H$ be a Hilbert space with an orthonormal basis $\{ x_n \}_{n=1}^\infty$ and let $\{ y_n \}_{n=1}^\infty$ be a sequence of elements in $H$. Show that following two statements are equivalent $$ \...
1
vote
0answers
23 views

Prove multiplication by sequence is a compact operator

Let $c_0(\mathbb N)$ be the space of sequence in $\mathbb C$ whose limit is zero, equipped with the $\ell^\infty$ norm. Let $u_n$ be a sequence in $\mathbb C$ and define the operator $A$ taking a ...
1
vote
0answers
19 views

$ (k\otimes h^\ast)^\ast=h\otimes k^\ast$?

Let $H,K$ be Hilbert spaces with $h\in H,k\in K$. Let $k\otimes h^\ast(g)= \left\langle g,h \right\rangle k$. I'm supposed to prove $ (k\otimes h^\ast)^\ast=h\otimes k^\ast$, but I don't see how this ...
1
vote
0answers
17 views

For which $\alpha$ is this integral operator compact?

I have $Q\subset\mathbb{R}^n$ $Af(x)=\int_QK(x,y)f(y)dy$ , with $K(x,y)=\frac{K_0(x,y)}{|x-y|^\alpha}$ and $K_0\in C(Q)$ I want to estimate using an operator $A_Mf(x)=\int_QK_M(x,y)f(y)dy$ where, $...
0
votes
1answer
18 views

Time (only) dependence with respect to the inner product of a wave function in $L^2(\mathbb{R})$

In my book "Quantum Theory for Mathematians" By B. Hall there is a discussion about the derivative of the inner product of a time-dependent wave functions $\psi(t)$ (note: no position dependence is ...
1
vote
0answers
23 views

Dimension and separability of $\ell^2(I)$?

Let $\ell^2(I)= \left\{ x:I\rightarrow \mathbb C\mid \sum _{i\in I} |x_i|^2<\infty \right\} $ with inner product $\sum_{i\in I}x_i \bar y_i$. I am supposed to find the dimension of this space and ...
2
votes
1answer
48 views

Weak convergence of product in $L^2$

Let $f_k\in L^\infty[0,1]$ and $g_k\in L^2[0,1]$ be two sequences such that $f_k\to f$ a.e., $\left\|f_k\right\|_{L^\infty}\leq c$ for any $k$ and $g_k\rightarrow g$ weakly in $L^2[0,1]$. Why does $...
0
votes
0answers
24 views

Does the Parseval identity imply the completeness of an orthonormal system?

Let $V$ be an inner product space which is not complete. Can there by an orthonormal system satisfying the Parseval identity for each vector but which is not complete i.e the only vector orthogonal to ...
1
vote
1answer
35 views

Prove $\exists x$ such that $\|Ax-b\|$ is minimal and this is unique if $A$ is invertible

Suppose $A\in \mathbb R^{m\times n}$. I'm supposed to prove $\exists x$ such that $\|Ax-b\|$ in minimal and this $x$ is unique if $A$ is invertible, in which case I also need to exhibit a formula. I ...
0
votes
1answer
16 views

Metric projection onto nonnegative sequences

I need to find the metric projection onto non-negative sequences in $\ell^2$. Intuitively I'm thinking each element in the sequence should be sent to the maximum between it and zero, since that should ...
1
vote
1answer
30 views

If $L$ is a continuous linear form on a dense subspace of a Hilbert space $H$, what do we mean by the claim $L\in H$?

Let $H$ be a $\mathbb R$-Hilbert space $D(\mathfrak a)$ be a dense subspace of $H$ $\mathfrak a:D(\mathfrak a)\times D(\mathfrak a)\to\mathbb R$ be bounded, i.e. $\exists c\ge 0$ with $$\left|\...
2
votes
1answer
28 views

Integral operator convergence study in $L^2(\mathbb{R})$

Exercise I want to study in $\mathcal{H}=L^2(\mathbb{R})$ the convergence of $$A_nf(x)=\log n \int_\mathbb{R} \frac{1}{1+n(x-y)^2}f(y)dy. $$ Solving a) Pointwise convergence : For $n \to \infty$ ...
1
vote
1answer
29 views

Verifying reproducing property of common kernels in RKHS

Consider a reproducing kernel Hilbert space $\mathcal{H}$, whose elements are functions from $X \rightarrow \mathbb{R}$. Let $t \in X$. Since $\mathcal{H}$ is a RHKS, there exists a unique function $...
0
votes
1answer
31 views

Smooth Approximations in $L^2((0,1))$

Let $L^2((0,1))$ be as usual the Lebesgue space of measurable complex-valued functions $f:(0,1) \rightarrow \mathbb{C}$ such that $\int |f(x)|^2 dx < \infty$. It is a well known fact (see e.g Lieb ...
1
vote
0answers
14 views

How is this function a member of $L^{1}(0, \frac{1}{b})$?

The function in question is: $$ F_{n}(x) = \sum_{k \in \mathbb{Z}} f\left(x-\frac{k}{b}\right) g^{\ast}\left(x - na - \frac{k}{b}\right) $$ Where $\ast$ denotes complex conjugation, $f, g \in L^{2}$,...
2
votes
0answers
23 views

Convolution is continuous

Convolution is continuous Let $f,g\in L^2\left(\mathbb T,\mathbb C\right)$ (Hilbert space of $1$-periodic functions) then $f*g$ should be continuous by Young's inequality (the map is $(f,g)\mapsto ...
1
vote
1answer
29 views

Confused about Domain of Unbounded Operators (Hilbert Spaces)

I understand that a bounded (linear) operator $A$ on a Hilbert space $H$ satisfies a condition $||Av|| \le c ||v|| $ for some fixed real number $c$ and for all $v \in H$. So, the domain of a bounded ...
0
votes
0answers
26 views

does an isometry preserve orthonomal basis?

I've got the following question. Let $\{\phi_n\}$ be any orthonormal basis of the reproducing kernel Hilbert space $H_0$. Let $F:H_0\to L^2$ be a linear isometry. Is $\xi_n=F(\phi_n)$ an ...
3
votes
2answers
63 views

Relationship between $C_c^\infty(\Omega,\mathbb R^d)'$ and $H_0^1(\Omega,\mathbb R^d)'$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the inner product on $L^2(\Omega,\mathbb R^d)$ $\mathcal D:=C_c^\infty(\Omega,\mathbb R^d)$ and $$...
0
votes
0answers
10 views

Property of the orthogonal projection $\tilde{\operatorname P}$ in the definition of the Stokes operator

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ $\langle\;\cdot\;,\;\cdot\;\rangle_2:=L^2(\Omega,\mathbb R^d)$ and $$\langle u,v\rangle_{1,\:2}:=\langle u,v\rangle_2+\sum_{i=1}^d\langle\nabla u_i,\...
0
votes
1answer
23 views

If $H$ is a Hilbert space, $U≤H$ is closed and $E≤U^⊥$ such that $x∈H$ with $x⊥_H E$ implies $x∈U$, then $U^⊥=\overline E^{\langle\;⋅\;,\;⋅\;\rangle}$

Let $\left(H,\langle\;\cdot\;,\;\cdot\;\rangle\right)$ be a separable Hilbert space $U$ be a closed subspace of $H$ $E$ be a subspace of $$U^\perp:=\left\{x\in H:\langle x,u\rangle=0\text{ for all }...
-1
votes
0answers
20 views

Prove this space is a Hilbert space

Let us consider a convex polygonal and bounded domain $\Omega$ in $\mathbb{R}^2$ containing two subdomains $\Omega_1, \Omega_2$ which satisfy $\overline{\Omega}=\overline{\Omega}_1\cup\overline{\Omega}...
0
votes
0answers
22 views

Characterization of the Gradient of a Distribution

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ (without topology) $u:\mathcal D(\Omega)\to\mathbb R$ is called distribution on $\Omega$ $:\...
2
votes
1answer
32 views

Is $p \vee q \leq p+q$ for $p,q$ projections?

I am wondering if $p \vee q \leq p+q$ for $p,q$ projections acting on some Hilbert space $H$. In particular, I wonder if the set of finite trace projections is upwards directed with the usual ...
0
votes
1answer
19 views

Span of Hilbert base

Let $\{e_j\}$ be an (infinite) Hilbert base of a Hilbert space $H$. Is the subspace $U=span_{\mathbb{C}}\{e_j|j\ge 0\}$ again a Hilbert Space? Thanks
0
votes
2answers
41 views

The form of a normal operator with only one element in its spectrum

Let be $H$ a Hilbert space. Show that if $T$ is a normal linear operator continuous (i.e. $T^*T = TT^*$, with $T^*$ the Hilbert adjunct of $T$) and your spectrum $\sigma(T) = \{\lambda\}$, than $T = \...
0
votes
0answers
18 views

Trace class norm and rank inequality

I am quite new to operators in Hilbert spaces and I have been trying to show that for any linear and bounded operator $T : \mathcal{H} \rightarrow \mathcal{H}$ \begin{equation} \vert \vert T \vert \...
2
votes
0answers
40 views

$W_0^{1,\:p}(\Lambda)$ is dense in $L^2(\Lambda)$

Let $d\in\mathbb N$ $\lambda$ be the Lebesgue measure on $\mathbb R^d$ $\Lambda\subseteq\mathbb R^d$ be open with $\lambda(\Lambda)<\infty$ $p\ge 2$ $W^1(\Lambda)$ denote the set of weakly ...