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

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5
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2answers
368 views

Optimization problem using Reproducing Kernel Hilbert Spaces

I am encountering a problem concerning Reproducing Kernel Hilbert Spaces (RKHS) in the context of machine learning using Support Vector Machines (SVMs). With refernece to this paper [Olivier ...
2
votes
0answers
98 views

Functions in a Reproducing Kernel Hilbert Space are Lipschitz continuous

I would like to show that all the functions in a Reproducing Kernel Hilbert Space (RKHS) are Lipschitz continuous. So that, I take two points in the domain $\vec{x}_{1} ,\vec{x}_{2} \in X$ then from ...
3
votes
1answer
72 views

Closure of an Operator in $l^2$

Let $l^2$ denote the Hilbert space of all complex sequences $\phi = (\phi_j)_{j=0}^{\infty}$ such that $\sum_{j=0}^{\infty} |\phi_j|^2 < \infty$. Consider the linear subspace of $l^2$ defined by ...
0
votes
0answers
8 views

Continuous inclusions in Hilbert-Sobolev space $H^s(\mathbb{R}^n) \hookrightarrow \mathcal{E}^k(\mathbb{R}^n)$

I have to prove that $H^s(\mathbb{R}^n) \hookrightarrow \mathcal{E}^k(\mathbb{R}^n)$ with $s \in \mathbb{R}$, $k \in \mathbb{N}$ and $s-k > n/2$, where $\mathcal{E}^k(\mathbb{R}^n):=\lbrace u: ...
1
vote
1answer
57 views

Parturbations of orthonormal bases [on hold]

Suppose that $(e_n)_{n=1}^{\infty}$ is an orthonormal basis in a Hilbert space $H$, and let $(f_n)$ be an orthonormal sequence in $H$ such that $$\sum_{n=1}^\infty \|e_n-f_n\|<\infty.$$ How can ...
1
vote
1answer
25 views

Finding distance of $h(t)=t$ from a closed subspace $Y$ of $\pi$-periodic functions in $L^2(-\pi,\pi)$

Let $Y=\{f\in L^2(-\pi,\pi):f(t-\pi)=f(t) \text{for almost all $t\in(0,\pi)$} \}$ Show that there exists $g\in Y$ such that $$\|h-g\|_2=\inf \{\|h-f\|_2:f\in Y\}$$ where $h(t)=t$. Compute ...
0
votes
0answers
7 views

Definition of essential spectrum?

Suppose we have a Hilbert space $\mathscr{H}$ and a bounded linear map $T\in\mathscr{B(H)}$ NOT necessarily self-adjoint. There seems to be loads of definitions of the essential spectrum of $T$. My ...
2
votes
1answer
60 views

Does $AB=(AB)^{\ast}$ and $A=A^{\ast}$ implies $B=B^{\ast}$?

Suppose that we have $AB=(AB)^{\ast}$ and $A=A^{\ast}$, does this implies that $B=B^{\ast}$? ($A^{\ast}$ is the Hermitian adjoint of $A$.) I have a feeling that they might not be equal in general. ...
2
votes
1answer
47 views

Explanation of “weight function” of inner product in Hilbert space

I am a physicist so I am sorry if the following is not written in a rigorous(or even completely right) way. As Quantum Mechanics is formed in Hilbert space, I would like to know what the weight ...
1
vote
0answers
19 views

Function with both easy to find Fourier and Hermitian coefficient

I'm writing some notes on Spectral theory and I would like to make a simple example finding the generalized fourier coefficient of a function in respect of two different bases. I was thinking about ...
0
votes
0answers
10 views

Norm and Inner Product Inequality in Hilbert spaces

Let $H$ be a Hilbert space, and suppose that $C \subset H$ is closed, convex and nonempty. Then, for $y_{j}=P_{C}(x_{j})$, $j=1,2$ where $P_{C}$ is the metric projection onto $C$ and $x_{1},x_{2} \in ...
0
votes
0answers
24 views

Distance from image of bounded operator

Let $A:H\rightarrow H$ be a bounded linear operator on Hilbert space $H$. Suppose we have $x\in H$ and $r>\mathrm{dist}(x,A(H))=\inf\{\|x-Ah\|,\ h\in H\}$. How to prove that then there exist ...
0
votes
3answers
172 views

show that $l^2$ is a Hilbert space

Let $l^2$ be the space of square summable sequences with the inner product $\langle x,y\rangle=\sum_\limits{i=1}^\infty x_iy_i$. (a) show that $l^2$ is H Hilbert space. To show that it's a ...
1
vote
0answers
48 views

Properties of weakly convergent series in Hilbert space

Let $H$ be a Hilbert space and $\{x_n\}_{n=1}^{\infty}$ given sequence of vectors from $H$. Suppose that for every $\{\alpha\}_{n=1}^{\infty}\in \ell^2$ series $\sum_{n=0}^{\infty}\alpha_nx_n$ is ...
0
votes
1answer
23 views

What does it mean for an inner product to be conjugate linear in the second entry?

Let $G$ be a group and $L^2(G) = \{f: G \rightarrow \mathbb{C} \}$. Now define an inner product on $L^2(G)$ by $$\langle f, g \rangle = \sum_{x \in G}f(x)\overline{g(x)}$$ Where $\overline{g(x)}$ is ...
2
votes
1answer
41 views

Is there a nice way to express $\psi_1$ using this orthonormal sequence?

Suppose that $H$ is a separable Hilbert space and $(\psi_n)_{n=1}^{\infty}$ is a complete orthonormal sequence in $H$. We define a sequence $(\phi_n)_{n=1}^{\infty}$ by $$ \phi_n=\psi_1+\psi_{n+1}\ ;\ ...
0
votes
0answers
25 views

Use Cauchy-Schwarz inequality to prove that $\langle\,,\rangle : \mathscr H \times \mathscr H \to \Bbb C$ is continuous.

Let $(a,b) \in \mathscr H \times \mathscr H$ be fixed. So we have to prove that for a given $\epsilon \gt 0$, we can find $\delta_1 \gt 0$ and $\delta_2 \gt 0$ such that $\lvert \langle x,y\rangle - ...
2
votes
4answers
50 views

Compact operator in Hilbert spaces reach the maximum in the sphere.

I found the following question in my textbook: (QUESTION) Let $\mathcal{H}$ a Hilbert space and $T: \mathcal{H} \rightarrow \mathcal{H}$ a compact operator. Show that exists $x \neq 0$ in ...
0
votes
0answers
14 views

Prove that a function is continuous (square integrability)

I need help for the following proof of continuity: Let $E=L_2([t_0,t_1],\mathbb R)$ be a Hilbert space of square-integrable real-valued functions on $[t_0,t_1]$. Let ...
0
votes
0answers
20 views

orthonormalization for a hilbert space

sincerely, I'm stuck. Then, I have two questions: if we take $V=\{v\in H^1(0,1) ; v(0)=0\}$ and $Q=\{ w_1,w_2\}$ is a lineary independent set where $w_1 = \frac{*}{\Vert *\Vert_{V\cap H^2(0,1)}}$ and ...
1
vote
1answer
23 views

If $U$ is unitary operator then spectrum $\sigma(U)$ is inside the unit circle- verification

In a Hilbert space, let $U$ be a continuous operator which it unitary. Prove $\sigma (U)\subseteq \Bbb{S}^1$. It is important for me to know how I am doing, and I didn't come by a clear explanation ...
1
vote
1answer
35 views

Tensor Product on Hilbert Spaces (well definedness)

Let $H_1$, $H_2$,...,$H_n$ be $n$ Hilbert Spaces. For each $\phi_i \in H_i$, Let $$ \phi_1 \otimes \phi_2 \otimes... \otimes \phi_n:= \text{Conjugate multilinear form which acts on $H_1 \times H_2.. ...
1
vote
0answers
22 views

Uncountable basis in Hilbert space

For a (uncountable dimension) Hilbert space $\mathcal{H}$, suppose we have uncountably many vectors $K_x$, only $0$ is orthogonal to all of $K_x$. (Specifically, a reproducing kernel Hilbert space ...
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0answers
20 views

Books to learn tensor product on hilbert spaces

I have just started to work on Quantum Computing. I have began to read a paper which deals with tensor product on hilbert spaces. I have a had a course in functional analysis. So I don't have an ...
1
vote
1answer
20 views

Form of the Polar decomposition for $M_{\varphi}$

Polar Decomposition:Let ‎$‎‎v$ ‎be a‎ ‎continuous ‎linear ‎operator ‎on a‎ ‎Hilbert ‎space ‎‎$‎‎H$.then ‎there ‎is a‎ ‎uniqe ‎partial ‎isometry ‎‎$‎‎u\in B(H)$ ‎such ‎‎$‎‎v=u‎‎\mid ‎v‎\mid‎‎‎$ ‎and ...
0
votes
1answer
20 views

Compact operator with no non-zero eigenvalues is zero?

Suppose we have a Hilbert space $H$ and a compact operator $T$ acting on $H$. If $T$ has no non-zero-eigenvalues, is it necessarily the zero operator? Secondly, if I decompose $H$ into eigenspaces of ...
3
votes
1answer
693 views

Hilbert cube is compact

Let $\{u_n\}_{n\in \mathbb N}$ be an orthonormal set in $H$ (Hilbert space). How prove that the set $\displaystyle Q=\{x\in H :\ x=\sum_{i=1}^{\infty}{c_nu_n}, \ \mbox{where} |c_n|\leq\frac{1}{n} \}$ ...
3
votes
1answer
43 views

Example of a self-adjoint bounded operator on a Hilbert space with empty point spectrum

Good day, I wanted to find a self-adjoint bounded operator on a Hilbert space with empty point spectrum i.e. $$ T = T^* ~\text{but}~ \sigma_p(T)= \emptyset $$ Some definitions and results of the ...
1
vote
0answers
21 views

How should $A^α$ be defined for real $α ∈ [0,∞)$ and $A\in M_n(\mathbb C)$?

Let $A\in M_n(\mathbb C)$ be arbitrary. I'm interested to know How should $A^{\alpha}$ be defined for real $\alpha\in [0,\infty)$? When $A$ is nonsingular, we can define $A^{\alpha}=\exp(\alpha ...
0
votes
2answers
57 views

Adjoint of an Operator in $l^2$

Let $l^2$ be the Hilbert space of all complex sequences $\phi =(\phi_j)_{j=0}^{\infty}$ such that $\sum_{j=0}^{\infty} |\phi_j |^2 < \infty$. Set $D= \{ \phi \in l^2 : \sum_{j=0}^{\infty} j ...
2
votes
1answer
22 views

Image of a dense set through unbounded operator

Let $T$ be a densely defined, closed operator on a Hilbert space $H$ such that $T^*T$ remains densely defined. Obviously, $\sigma(I+T^*T)\subset [1,\infty)$, which in particular implies this operator ...
0
votes
0answers
27 views

Mercer's expansion on Sinc function

I hope to know about the Mercer's expansion on $K(x,y) = \frac{\sin(x-y)}{\pi(x-y)}$, which is the reproducing kernel for a Hilbert space of band-limited functions. By Mercer's theorem, it can be ...
0
votes
0answers
26 views

On the sequence of orthonormal basic

I have a question : Let $0 \leq a \leq b \leq +\infty $, supposing that ${\phi_n(x,t)}_{n \geq 0}$ be the orthonormal basis on $L^2(a,b)$ respected to $x$. If there exist a sequence $\psi_n(t)$ such ...
0
votes
1answer
36 views

Characterization Projection operator as distance minimizer

Let $H$ be a Hilbert space and $V$ be a subspace of $H$. How can I prove that for a map $P \colon H \rightarrow V$ the following are equivalent: $P^2=P$ and $P$ is linear $P(x) = ...
45
votes
3answers
5k views

Connections between metrics, norms and scalar products (for understanding e.g. Banach and Hilbert spaces)

I am trying to understand the differences between $$ \begin{array}{|l|l|l|} \textbf{vector space} & \textbf{general} & \textbf{+ completeness}\\\hline \text{metric}& \text{metric ...
3
votes
1answer
77 views

Normal Operators: Superalgebra (I)

Given a Hilbert space $\mathcal{H}$. Consider normal operators: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ Denote their calculus: ...
1
vote
1answer
100 views

Show $\Bigl\{\sqrt{2\over {\pi}}\sin (nx)\Bigr\}_{n=1}^{\infty}$ is an orthogonal basis of $L_2[0,\pi]$

Show $\Bigl\{\sqrt{2\over {\pi}}\sin (nx)\Bigr\}_{n=1}^{\infty}$ is an orthogonal basis of $L_2[0,\pi]$. What I need is a verification and guidance. I managed to show that the set is orthogonal. My ...
1
vote
0answers
13 views

Find the codimension of $\overline{\operatorname{span}}\{S^n(1,4,-1,0,0,\ldots):n=0,1,2,\ldots\}$ in $\ell_2$.

Find the codimension of $A=\overline{\operatorname{span}}\{S^n(1,4,-1,0,0,\ldots):n=0,1,2,\ldots\}$ in $l_2$ where $S$ is the shifting operator to the right: $Se_i=e_{i+1}$. I don't quote understand ...
-1
votes
0answers
15 views

Is the distance attained?

Suppose that we consider the set $K:=\{ x \in \mathbb{R}^n: \sum_{j=1}^n |x_j|^p \leq 1 \}$ where $0<p<1$. In this case the set isn't convex. Indeed, if we pick for example $x=(1,0,0, \dots), ...
2
votes
1answer
802 views

Sesquilinear forms on Hilbert spaces

Definition of the problem Let $\mathcal{H}$ be a Hilbert space, and let $B:\mathcal{H}\times\mathcal{H}\rightarrow\mathbb{K}$ be a sesquilinear form. Prove that TFAE: $(i)$ $B$ is continuous. ...
4
votes
0answers
55 views

Null Functional on $l^2$

Let $l^2$ be the hilbert space of all complex sequences $\psi= (\psi_n)_{n=0}^{\infty}$ such that $\sum_{j=0}^{\infty} |\psi_j |^2 < \infty$. Let $\phi= (\phi_n)_{n=0}^{\infty}$ be a sequence of ...
1
vote
1answer
34 views

Open sets in the unitary group $ U(\mathcal{H}) $ of a Hilbert space $ \mathcal{H} $.

Let $H$ be an infinite dimensional Hilbert space and let $(x_i)_1^\infty$ be an orthonormal basis for $H$. Consider $U(H)$ the unitary group of the continuous unitary operators on $H$. Equip $U(H)$ ...
0
votes
1answer
21 views

A question concerning the triplet $V\subset H\subset V^*$

In Brezis' Functional Analysis book, p. 150, there is an exercise about the triplet $V\subset H\subset V^*$, where $(V,\|\cdot\|_{V})$ is a Banach space, $H$ is a Hilbert space with the scalar product ...
1
vote
1answer
27 views

Find Riesz representation of $\phi=f({1\over 2})$

"Let $\rho$ be a space of complex polynomial and define $<f,g>={1\over 2\pi}\int_{0}^{2\pi}f(e^{it})\overline{g(e^{it})}dt$ for $f,g:\rho\to \Bbb{C}$. Let $\phi$ be a linear functional on ...
3
votes
0answers
49 views

Proving compactness of an operator $(Kf)(t)=\int_{0}^{\infty}k(t+s)f(s)ds$

I was trying to prove the compactness of the following operator: $K:L_2([0,\infty))\to L_2([0,\infty))$ $(Kf)(t) = \int_{0}^{\infty}k(t+s)f(s)ds$, given that the function $k$ is continous, and ...
1
vote
1answer
481 views

distributivity of tensor product and direct sum for Hilbert spaces

Before I ask my actual question about direct sums and tensor products of Hilbert spaces, let's first talk about direct sums and tensor products of vector spaces. We might define direct sums of ...
0
votes
2answers
56 views

problem on hilbert spaces

Let $X=C[0,1]$ with the inner product $\langle x,y\rangle=\int_0^1 x(t)\overline y(t)\,dt$ $\forall$ $x(t),y(t)\in C[0,1]$ $X_0 =\{x(t) \in X :\int_0^1 t^2x(t)\,dt=0\}$and $X_0^\bot$ be the ...
3
votes
1answer
52 views

Bounded Operators on a finite-dimensional Hilbert space - Linear combination of at most two unitaries and from a partial isometry to a unitary

Good day, In the lecture of functional analysis the proof of two statements were skipped as a task for us but I'm not sure how I approach these questions. a) Show that every partial isometry $V \in ...
1
vote
0answers
40 views

Extend and restriction of operator on $B(H)$

Let ‎$‎‎H$ ‎be a ‎Hilbert ‎space ,‎‎‎‎‎‎$‎‎B(H)$ ‎be ‎bounded ‎operators ‎on ‎‎$‎‎H$ ‎and ‎‎$‎‎K(H)$ ‎be ‎compact ‎operators ‎on ‎‎$‎‎H$‎. Assume ‎that ‎‎$‎‎M$ ‎is a ‎close‎d subspace of ‎$‎‎H$ ‎and ...
0
votes
0answers
16 views

Compute the limit and show that uN converges weakly

full question I already know that the norm is 1, and that you can use the definition of weak convergence but that's where I get lost. Somebody told me I can use the Riesz representation theorem since ...