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

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37 views

How to prove this specific kernel is not in RKHS?

Consider $\mathcal{X}=\mathbb{R}$, and $k(x,y)=xy=[\frac{x}{\sqrt{2}},\frac{x}{\sqrt{2}}]\cdot [\frac{y}{\sqrt{2}},\frac{y}{\sqrt{2}}]^T$, where we thus can define two kinds of feature maps ...
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42 views

How to prove $\Phi_w(x)=\cos (w^T x+b)$ is outside the RKHS associated with the Gaussian kernel function$K(x,y)=\exp(-\frac{||x-y||^2}{2\sigma})$?

How to prove $\Phi_w (x)=\cos (w^T x+b)$ is outside the RKHS ( Reproducing Kernel Hilbert Space) associated with the Gaussian kernel function$K(x,y)=\exp(-\frac{||x-y||^2}{2\sigma})$?
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45 views

How to understand ' Let $\mathcal{H}$ be a Hilbert space of functions $f$ : $ \mathcal{X} \rightarrow R$, denoted on a non-empty set $\mathcal{X}$.'

I am a beginner. By asking this question, I means that, to construct a Hilbert space, should $\mathcal{X}$ satisfy some properties? Furthermore, in some papers especially on machine learning, ...
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1answer
69 views

Why $\|f\|_2^2$ can be written as $\int |f(x)|^2 p(x)dx$?

I am recently learning some papers on optimization in infinite-dimensional space, and I not familiar with function analysis. In some papers, I see $\|f\|_2^2$ is written as $\int |f(x)|^2 p(x)dx$ , ...
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2answers
58 views

What is the definition of closed subspace?

I am trying to understand what is intended with closed subspace, I took the following guess: A closed subspace $M$ of a Hilbert space $H$ is a subspace of $H$ s.t. any sequence $\{x_n\}$ of elements ...
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1answer
71 views

$L_2$ norm and RHKS norm in Hilbert spaces $\mathcal{H}$

According to this paper (just right below the Theorem 3 and above the section 3) Reproducing Kernel Hilbert Space(RKHS) $\mathcal{H}$ on $\mathcal{X}$ is a Hilbert space of functions from ...
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1answer
58 views

The inner product of the Cartesian Product space

I want to know how can one define the inner product in the Cartesian product of spaces, i.e. let $A,B$ two hilbert spaces. Let $a_1, a_2 \in A$ and $b_1, b_2 \in B$, how can one express the inner ...
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3answers
63 views

On the intuition behind the projection theorem.

I have recently proved the projection theorem in a Hilbert space setting. The statements were: If $M$ is a closed subspace of a Hilbert space $H$ and $x \in H$, then: There is a unique element ...
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1answer
27 views

Determining the orthogonal complement of $\{1 \}^\perp$ in $L^2[0,1]$

Consider the space $L^2[0,1]$ of complex valued square-integrable functions $f : [0,1] \to \mathbb{C}$. Let $\langle f, g \rangle = \int_0^1 f \bar{g}$ denote the standard $L^2$ inner product. For $M ...
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1answer
23 views

The Spectral Radius of a Product of Two Hilbert-Space Operators

I’m given a Hilbert space $ \mathcal{H} $ such that $ \dim(\mathcal{H}) > 1 $, and I’m supposed to construct two operators $ A $ and $ B $ on $ \mathcal{H} $ such that $ r(A B) \neq r(A) r(B) $. Is ...
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1answer
128 views

What is the orthogonal of an intersection?

Background I have been introduced to the notion of orthogonal complement of a subset of a (pre)hilbert space. Given $X$ a (pre)hilbert space and $A\subseteq X$, one defines $A^\perp:=\{x\in X:x\perp ...
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60 views

Construct a unitary operator U on H with prescribed spectrum

Given an infinite dimensional Hilbert space $H$. Let $|\lambda_k| = 1$ for $k = 1, ..., n$. Construct a unitary operator $U$ on $H$ such that $\sigma(U) = \{\lambda_k\}$ for $k=1,....,n.$ I can ...
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2answers
35 views

Construct a bounded linear operator S on H such that σ(S) = A

Given an infinite dimensional Hilbert space $H$. Let $A\subseteq \mathbb{C}$ be closed and bounded. Construct a bounded linear operator $S$ on $H$ such that $\sigma(S)=A$, where $\sigma(S)$ is the ...
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1answer
17 views

On the polar representation of an inner product.

Take $H$ an inner product space. $x,y \in H$. Take $b = |<x,y>|$ . Then the polar representation of $<x,y>$ is: $$<x,y> = be^{i\theta}$$ for some $\theta \in (-\pi, \pi]$. Why is ...
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1answer
48 views

Show that a subspace is closed in Hilbert space $H$

Let $u\in B(H)$ , $\lambda < 0$. Also we have $\|(u-\lambda)x\|\geq |\lambda|\|x\|$. So $u-\lambda$ is bounded below. To show $(u-\lambda)(H)$ is closed in $H$, suppose $\{(u-\lambda)x_n\}$ be ...
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1answer
49 views

What is the 'largest' space of integrable functions which is also a Hilbert space?

It is well known that $L^2(X,\mu)$, the set of functions $f:X \rightarrow \mathbb{C}$ such that $\int_X |f|^2 \text{d} \mu < \infty$, is a Hilbert space. Is there a Hilbert space $H$ such that ...
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37 views

Basis of $H^1(\Omega)$ which is orthonormal wrt. $L^2(\partial\Omega)$ inner product?

Let $\Omega$ be a domain with $\partial\Omega$ bounded. Is it possible to find a smooth basis of $H^1(\Omega)$ and $L^2(\Omega)$ which is orthonormal wrt. the $L^2(\partial\Omega)$ inner product? ...
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86 views

Approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets?

Are there some common ways to approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets? (note that the unit ball isn't compact.) My goal is to prove a statement which holds ...
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1answer
33 views

For $(l^2,\|\cdot\|_2)$ and $e_n=(0,0,.,1,0,.)$ and a bounded linear functional $\Phi$ find $p\geq 1$ where $\sum_{n=1}^\infty |b_n|^p$ converges?

For $(l^2,\|\cdot\|_2)$ and $e_n=(0,0,...,1,0,...)$ and a bounded linear functional $\Phi$ find a value of $p\geq 1$ where $\sum_{n=1}^\infty |b_n|^p$ converges for $b_n=\Phi(e_n)$? Ok so since ...
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90 views

If $H$ is a one-dimensional Hilbert space then the zero representation of a C*-algebra on $H$ is irreducible.

It says on page 143 of Murphy's $C^*$-algebras and operator theory that if $H$ is a one-dimensional Hilbert space then the zero representation of any C*-algebra on H is irreducible. What is the zero ...
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49 views

the norm of a linear operator

In in the demonstration of Lax-Milgramm lemma, they use a linear operator $A:V\to V$, where $V$ is a Hilbert space; My basic problem is how to prove that $$\|Au\|_V=\sup_{v\in ...
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1answer
42 views

Am I doing this approximation correctly? (least squares method)

Here is the problem. Find the function $f$ of the type $f(x) = a\cos x + b\sin x$ which best approximates the function $g$ in the points : $$ \begin{array}{ c | c | c | c | c | c | c } x & ...
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2answers
29 views

Terminology for orthogonal projections

Let $H = X \oplus Y$ a Hilbert space. Then, the map $p(x + y) = x$ is called the orthogonal projection onto $X$ along $Y$. Why is it necessary to mention along $Y$? Of course if a space has a ...
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1answer
39 views

Norm of an operator and eigenvalues

I have $K\colon L^2(0,T) \to L^2(0,T)$ a Hilbert-Schmidt integral operator (and so $K$ is linear, bounded, compact and self-adjoint) and I have obtained its eigenvalues and eigenvectors. From them, I ...
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28 views

Spectral Measures: Completeness

Given a Borel space $\Omega$ and a Hilbert space $\mathcal{H}$. Consider a spectral measure $E:\mathcal{B}(\Omega)\to\mathcal{B}(\mathcal{H})$. A spectral measure can be completed $\overline{E}$. ...
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1answer
55 views

How to show that the operator $T(\{x_n\})=\{n x_n\}$ has closed graph?

Consider the subspace $$D=\left\{x\in \ell^2 \ \big|\ \sum_{n\in\mathbb N} n^2 |x_n|^2<\infty\right\}$$ of $\ell^2$, and let $T:D\to\ell^2$ be defined by $T(\{x_n\})=\{n x_n\}$. I need ...
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43 views

Fréchet derivatives of $\sum_{n=1}^\infty x_n^2/n^3 -\sum_{n=1}^\infty x_n^4$

I read that the second order Fréchet derivative $F''(0)$ of linear functional $F:\ell_2\to\ell_2$, where $\ell_2$ is the separable real Hilbert space, defined by ...
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1answer
23 views

Show that this operator is linear

Let $\Bbb H$ is a Hilbetr space and $T:\Bbb H\to\Bbb H$ be a operator such that $$<x,Ty>=<Tx,y>$$ $\forall x,y\in\Bbb H.$ I want to show that $T$ is linear and bounded. If I can show that ...
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91 views

Derivative of norm in Hilbert space

I read (p. 485 here) that the Fréchet derivative of norm (non-linear) functional $p:H\to\mathbb{R}$, $x\mapsto\|x\|$ is $\frac{x}{\|x\|}$ for all $x\ne 0$, which I think to be intended as the linear ...
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30 views

Sufficient conditions for weak continuity

Are there any "easily verifiable" sufficient conditions for weak (equivalently, weak*) continuity of (not necessailry linear) maps on the unit ball of $\ell^2$, mapping into $\ell^2$? Apologies for ...
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1answer
21 views

Subspace of a Hilbert space with a distinct inner product

I don't really know where to begin with the following question: Let $ (H_0, \langle \cdot \rangle_0)$ be a closed subspace of $ (H, \langle \cdot \rangle )$ such that norms induced by $ \langle \cdot ...
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1answer
26 views

Proving an orthogonal projection of the Hilbert adjoint is just the adjoint

I'm facing the following problem: let $ H_0 \subset H $ be a $ T$-invariant closed subspace of Hilbert space $ H $ (i.e. $ T(H_0) \subset H_0 $) and $ P$ - an orthogonal projection of $ H $ onto $ ...
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1answer
184 views

Eigenvalues of Left Shift + Right Shift in $l_2([0,\infty))$

This question appeared on an old final exam and I am having difficulty completing it for practice. Let $S_r$ and $S_l$ be defined on the hilbert space $l_2[0,\infty)\to l_2[0,\infty)$ as the ...
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1answer
27 views

If $E$, $\overline{E}$ are orthogonal projections such that $\mathrm{range}(\overline{E})=\overline{\mathrm{range}(E)}$, then is $E\ge\overline{E}$?

I feel like this should be true. Let $\mathrm{range}(E)=A$ and $u$ be an arbitrary vector in a Hilbert space $H$, it is sufficient to show $\langle (E-\overline{E})u,u\rangle=0$. By Cauchy-Schwartz: ...
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1answer
58 views

Finite rank approximation of bounded operators on Hilbert space

Let H be a (finite dimensional) Hilbert space. The approximation property states that every bounded operator from H to itself can be approximated by a sequence of finite rank operators. My question ...
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1answer
20 views

About the von Neumann decomposition

The von Neumann theorem states that for any symmetric operator $f$, the domain $D_{f^\dagger}$ of its adjoint $f^\dagger$ is the direct sum of the three subspaces $D_{\bar{f}}$, $\aleph_z$, and ...
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1answer
47 views

CAR- & Weyl-Algebra: Uniqueness?

Given a Hilbert space: $\mathcal{h}$ Consider representations of the CAR-algebra: $\mathcal{A}_\text{CAR}^{(\prime)}(\mathcal{h})$ In Bratelli & Robinson it is stated the uniqueness: ...
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23 views

continuos spectrum of $R+L$, where $R$ and $L$ are the right and left shift of sequences in $l_2$

consider the Hilbertspace $l^2$ and the Left and Right-shift operator \begin{align*} L(x_1,x_2,\cdots) &= (x_2,x_3,\cdots)\\ R(x_1,x_2,\cdots) &= (0,x_1,x_2,\cdots ) \end{align*} I'm trying ...
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36 views

Any example of non-closed operator?

I cannot think of one. By the way, is there any good exercise book on functional analysis or hilbert space?
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237 views

Unit sphere weakly dense in unit ball

I'm studying for an exam and came across a problem: I want to prove that the unit sphere in a Hilbert space $\mathcal{H}$ is weakly dense in the unit ball. I already had to prove that the unit ball ...
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1answer
113 views

Spectral Measures: Lebesgue

Preface Dominated convergence: $$f_n(\omega)\to f(\omega)\quad(\omega\in\Omega)\implies f_n(E)\to f(E)$$ (This gives a tool for analysis of operators.) Problem Given a Borel space $\Omega$ and a ...
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3answers
55 views

Why cannot a densely defined operator be extended to an everywhere defined operator?

I am a physicist learning functional analysis because of its fundamental role in quantum mechanics. There are so many bizarre facts. One is, there are densely defined operators which seem cannot be ...
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1answer
53 views

Are all Banach spaces also Hilbert spaces?

We have the well-known "polarization identity" $$(x,y)=\frac{1}{4}\left(\|x+y\|^2-\|x-y\|^2+i\|x+iy\|^2-i\|x-iy\|^2\right)\tag{1}$$ that works in any Hilbert space. Hence, is every Banach space also a ...
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22 views

Why only densely defined operators can have an adjoint operator?

Why is it impossible or making no sense to define an adjoint operator for a non-densely defined operator?
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275 views

An example of non-closed subspace of a Hilbert space?

I am reading a book on Hilbert space. It seems that the author assumes that a linear subspace of a Hilbert space can be non-closed. I cannot think of an example. I am still used to the ...
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42 views

Properties of Hilbert Spaces- Contrasting Two Different Topological Spaces

Let H be the space of real sequences x = $(x_1 , x_2, ... )$ with $\sum(x_n^2)$ finite. (This is $l_2$ in fact.) I wish to show the following: The topology on H is ...
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31 views

Measurability of inner integral $x \mapsto \int f(x,y)\, d\mu(y)$

Let $\psi$ be defined by$$\psi(s):=\int_{[a,b]}K(s,t)\varphi(t)d\mu_t$$ where $\varphi\in L_2[a,b]$ and $K\in L_2([a,b]^2)$. Kolmogorov-Fomin's proves the belonging of $\psi$ to $L_2[a,b]$ by showing ...
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1answer
85 views

Orthonormal Hamel Basis is equivalent to finite dimension

Consider a Hilbert space which is infinite dimensional. If it is separable, it is well known that an orthonormal basis will be countable, while a hamel basis will be uncountable (since it is a ...
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22 views

Let $H$ be a Hilbert space, $A$ is unitary and $S=\{Ax:x\in H\}$. Does $S^{\perp}=\operatorname{Null}(A)$?

Let $H$ be a Hilbert space, and $S=\{Ax:x\in H\}$. Does $S^{\perp}=\operatorname{Null}(A)$? What I have is if $x\in S^{\perp}$ then $x\perp A(A^*A^*Ax)$ then $(x,A^*Ax)=(Ax,Ax)=0$, so $x$ is in ...
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32 views

0 limit point of spectrum of completely continuous operator $H\to H$

I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа (p. 475 here) that 0 is an accumulation point for the spectrum of a completely continuous operator $A:H\to H$ where $A$ ...