Questions tagged [hilbert-spaces]
For questions involving Hilbert spaces, that is, complete normed spaces whose norm comes from an inner product.
8,501 questions
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Proving an equivalent form of the essential spectrum for a self adjoint densely operator.
Let $T$ be a self adjoint densely defined operator on a Hilbert space. I am trying to show that a real number $\lambda$ will be in the essential spectrum iff there exists an orthonormal sequence $e_n$ ...
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Bounding a quadratic form using projections
Let $\mathcal{H}$ be a Hilbert space and $T$ a positive linear operator, meaning that $\langle x, T x\rangle \ge 0$ holds for every $x \in \mathcal{H}$. Let $M\subset \mathcal{H}$ be a closed subspace,...
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ONB for an unbounded self-adjoint operator
Let $H$ be a Hilbert space and $D:\mathrm{Dom}(D) \to H$ a densely defined operator on $H$. We further assume that $D$ is closed and self-adjoint. Does it follow, like in the fin-dim case, that $H$ ...
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Existence of norm 1 vector in Hilbert space that gives distance via inner product
I am trying to prove the following:
Let $H$ be a Hilbert space, $Y\subset H$ a closed subspace, and $x \in
H \backslash Y$. Then there exists a unique $z\in Y^\perp$ with $\| z
\| =1$ and $\langle ...
- 388
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Prove that $tr(T)=\sum _{\lambda \in \sigma_p(T)}\lambda \dim E_\lambda$
$T$ is a compact operator and a Trace-class operator on the Hilbert space $H$, prove that $tr(T)=\sum _{\lambda \in \sigma_p(T)}\lambda \dim E_\lambda$, where $E_\lambda =\ker(\lambda I-T)$.
The ...
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How to prove that Trace-class operators are Hilbert-Schmidt operators
Prove that if $T$ is a Trace-class operator on the Hilbert space $H$, then $T$ is also a Hilbert-Schmidt operator.
The definitions of Trace-class operators and Hilbert-Schmidt operators are as in ...
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Unbounded operator on $L^2([a,b])$ with bounded inverse
Does there exist any unbounded operator $A$ on $Dom(A)\to L^2([a,b])$, with $Dom(A)\subseteq L^2([a,b])$ such that its range is dense (or onto) in $L^2([a,b])$ and it has an inverse defined on the ...
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Does the product of two reproducing kernels lie within the same RKHS?
Consider $k$ to be the reproducing kernel for the functions defined on the space $X\times Y$. Denote the $\mathcal{H}_{(X,Y)}$ as its RKHS. I would like to approximate a conditional expectation using ...
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Show that the set $\{(x_k \cdot 2^{-k}) \mid x \in \ell_2\}$ is dense in $\ell_2$
I want to show that the set $A := \{(x_k \cdot 2^{-k}) \mid (x_k) \in \ell_2\}$ is dense in $\ell_2$.
My first attempt was that for any $(y_k) \in \ell_2$ also the sequence $(z_k) := (2^m \cdot y_k)$ ...
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Linear functional on a Hilbert space defined by a sesquilinear form can be expressed in certain manner when restricting to a finite dim. subspace
Let $H$ be a Hilbert space and $a : H \times H \to \mathbb{K}$ where $\mathbb{K}$ is either complex or real numbers, a sesquilinear form such that there exist constants $C, c > 0$ such that
$$ |a(...
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Let T be a bijective bounded linear operator on a Hilbert space H. Assume $T^{-1}$ exists and is bounded. Show that $(T^*)^{-1}$ exists.
This is my current attempt to this problem:
$T^*$ is injective:
I want to show $\mathcal{N}(T*) = \{0\}.$
Consider $T^*(u)=0.$ Then $<T^*(u), v> = 0$ for each $v \in H.$ By the definition of $T^*...
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Why does f(-x + y) = f(x) + f(y) hold, if f(x) = <a, x>? [closed]
I'm confused with answer to the question I'm working on right now, and would be grateful if anyone could please help about it.
The question is:
Let H be a Hilbert Space. Let vector a be from H, and f(...
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Finite-rank operator convergence over separable Hilbert space
Suppose I have an operator $A$ on some separable Hilbert space $\mathcal{H}$, which might have infinite rank (the rank is defined as the dimension of the range).
In other words, we take an orthonormal ...
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Operators on a Hilbert space with equal spectrum
Suppose $T_0$ and $T_1$ are two bounded operators on a Hilbert space $H$ $(T_0,T_1\in B(H))$ with $\sigma(T_0) =\sigma(T_1)$.
I want to show that there exists a bijective $*$-homomorphism between $C^*(...
user1446697
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Finding orthonormal basis of eigenfunctions for integral operator
We consider the operator $K: L^2([0,1]) \rightarrow L^2([0,1])$, defined by
$$(Kf)(t) = \int_0^1 \cos(2 \pi (t-s)) f(s) ds.$$
As an exercise, I am asked to compute its eigenvalues and eigenfunctions. ...
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Understanding multidimensional quantum systems
I'm struggling to understand how to expand the formalism developed in class to multidimensional systems. I'm studying physics and we really didn't cover anything regarding separation of Hilbert spaces,...
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Distance from function to subspace in Hilbert space
Consider the space $H = L^2[0,1]$ and the subspace
$$W = \{f\in H \:|\: \int_{0}^{1/2} f(x)=0\}.$$
Find the distance of the function $x^2$ to the subspace $W$. First off, the distance is well defined ...
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Prove that $L^2=L^+ \oplus L^-$.
Let's consider $L^2(\mathbb{R})=\{f:\mathbb{R}\to \mathbb{R} \mid \int_{\mathbb{R}}|f|^2\ d\lambda<+\infty\}$ where $\lambda$ denotes the Lebesgue measure.
So, $L^2$ endowed with the standard inner ...
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Hilbert space equality problem
i have a question regarding one equality, namely,
$$\langle Az,z\rangle =0$$ Where $z\in range A^*=(kerA)^{\bot}$
A is bounded linear operator.
I don't see why should this dot product be zero, Any ...
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For a functional, why is a subspace $J$ ideal?
I'm reading "A Mathematical Introduction to Conformal Field Theory" (you can read here) and I'm trying to show a subset $J$ is an ideal(p140-p141). Following images include notations.
Here, ...
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Definition and properties of a transpose map on an arbitrary Hilbert space $H$.
I am given the following definition and remark in my book:
Definition:
Let $A$ be a $C^{*}$-algebra and $\phi: A \rightarrow B(H)$. We say that $\phi$ is copositive if $t \circ \phi$ is completely ...
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Problem understanding conditional expectations as a projection
Let $(\Omega, \mathcal{A}, P)$ be a (complete) probability space and let $\mathcal{F}$ be a sub-$\sigma$-algebra of $\mathcal{A}$. Consider the mapping $1: \Omega \rightarrow \mathbb{R}, \, \omega \...
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Is the closed unit ball in a RKHS closed in $ L^2$ norm?
Background and notations
Let $\mathcal{X}$ denote an input domain. Let $H$ denote the Reproducing Kernel Hilbert Space (RKHS) induced by the Radial basis function (RBF) kernel with bandwith 1, which ...
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If $\phi:A\to B$ is a positive map of $C^*$-algebras and $e$ a projection such that $\phi(e) = \phi(1)$ then $\phi(a) = \phi(eae)$ for all $a ≥ 0$.
I am trying to understand the proof of the following statement:
Let $\phi: A \rightarrow B$ be a positive map, $A$ and $B$ being $C^{*}$-algebras. If $e$ is an orthogonal projection in $A$ such that $\...
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Convexity of functions on a Hilbert space
I am having some trouble finding a good reference on convex optimization on Hilbert spaces.
Let $(H, \langle \cdot, \cdot \rangle)$ be a Hilbert space and $f : H \to \mathbb{R}$.
If $f$ is twice ...
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Exact form of the operator when its adjoint operator is given
I am studying the adjoint of an operator defined from a Banach space to another Banach space. Let $X$ and $Y$ be the Banach spaces. $X^*$ denotes the dual space of $X$. Let $T:X\to Y$ be a bounded ...
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Isometries and Isomorphisms on Hilbert spaces
I'm using Conways Functional Analysis book right now and I'm a bit confused about isometries and isomorphisms. I read online that an isometry between Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$ is ...
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dense subspace of a Hilbert space is the domain of some unbounded operator
The following is a statement from the text Non Homogeneous Boundary Value Problems and Applications by Lions and Magenes section 2.1. We aim to show that Given two Hilbert spaces $X\subset Y$ with $X$ ...
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Expectation out the inner product [closed]
We define on a Hilbert space $\mathcal{X}$ with respective Borel probability measures $p$ and $q$. Let $f$ and $g$ be a function in $\mathcal{X} \times \mathcal{X}$.
My question: Under what conditions ...
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How Does One Verify This Operator Norm Inequality?
I am looking at the following problem from a functional analysis book, and unsure how they arrive at this conclusion:
Let $H,L$ be Hilbert spaces and $A$ a $C^{*}$-algebra with $\pi_H: A \...
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Infinite orthonormal bases are not Hamel: can we prove this without using Fourier series?
There is a well-known fact that for a Hilbert space $H$ with a choice of an orthonormal basis $\Gamma \subset H$, we have $\dim H < +\infty$ if and only if $\operatorname{span} \Gamma = H$, and ...
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Definition of sum of a family of mutually orthogonal projections
Suppose that $(P_i)$ is a family of mutually orthogonal projection acting on a Hilbert space $H$. The family need not to be countable in general. What is the definition of $\sum_i P_i$? More precisely,...
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The Norm of A Particular Family of Infinite Nonnegative Matrices
Let $H$ be a separable infinite dimensional Hilbert space. Let $E=(e_n)$ be an orthonormal basis of $H.$ If for all $m,k\in\mathbb{N},$ $\langle ae_m,e_k\rangle\geq 0,$ $\sum_{j=1}^\infty\langle ae_m,...
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Technical term for a part of a tensor product of spaces.
Consider a Hilbert space $\mathcal{H}= \otimes_i \mathcal{H}_{i}$. Consider two factor spaces $\mathcal{H}_S,\mathcal{H}_F \in \{\mathcal{H}_1, \mathcal{H}_2, \dotsc ,\mathcal{H}_1 \otimes \mathcal{H}...
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Weak Banach vs Arens-Mackey Topology
I've heard that, for general Hilbert $H$, the Arens-Mackey topology is stronger than the weak Banach topology on $B(H).$ Is this true? I find it doubtful.
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Is a monotone and Lipschitz continuous operator weakly sequentially continuous?
Let $\mathcal{H}$ be a real Hilbert space. Let $F:\mathcal{H}\to\mathcal{H}$ be monotone, i.e.,
\begin{equation}
(\forall x,y\in\mathcal{H}) \quad \langle x-y, F(x) - F(y)\rangle\geq 0,
\end{equation}
...
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Infinite sums and closure of linear subspace generated by finite linear combinations of orthonormal set in Hilbert Space
If we have an orthonormal set $(\phi_k)_{k = 1}^{\infty}$ in a Hilbert Space H and we set $M = \overline{\text{lin}}(\phi_1, \phi_2, ...)$ (that is, the closure of the set of all finite linear ...
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$A\in L(\mathcal{X},\mathcal{Y}), \ker(A^*)=\{0\}\Rightarrow Ran(A)=\mathcal{Y}$
In Charles A. Micchelli and Massimiliano A. Pontil. 2005. On Learning Vector-Valued Functions. Neural Comput. 17, 1 (January 2005), 177–204, the authors design a linear, continuous application $A$ ...
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How Hilbert space is a topological space? [closed]
I'm struggling with definitions. I go from https://en.wikipedia.org/wiki/Hilbert_space:
A Hilbert space is a real or complex inner product space that is also
a complete metric space with respect to ...
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Is pre-unitary operator on Hilbert space unitary?
Ramakrishnan and Valenza define a pre-unitary operator on a pre-Hilbert space to be an operator that preserves the Hermitian form or, equivalently, that $T^*T=1$. (They actually write $TT^*=1$ but I ...
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Defining representation of locally compact group on a pre-Hilbert space
Ramakrishnan and Valenza's Fourier Analysis on Number Fields defines a topological representation of a locally compact group only on a locally convex topological vector space. But they also talk of ...
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$|ta+(1-t)b|^p\leq t|a|^p + (1-t)|b|^p - \frac{t(1-t)}{C}|a-b|^p$ for a Hilbert space and $p\geq 2$
Let $H$ be a Hilbert space. For any $p\geq 2, t\in (0,1)$ and $a,b\in H$, I would like to prove (or disprove) that there exists a constant $C=C(p,t)$ that can depend on $p,t$ such that
$\begin{...
- 1,653
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Uniqueness of representation for orthonormal basis in Hilbert Spaces
I'm studying Hilbert spaces for Functional Analysis and we've recently got to the definition of an orthonormal basis for the space H (infinite dimensional) consisting of $\{e_n\}_{n \in N} \in H$ such ...
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Proving all pure states on $\mathcal{B}(H)$ live inside the w* closure of the vector states
I'm reading Takesaki's Theory of Operator Algebras I and I'm stuck on one of the exercises. This is exercise 2 in Section 2 of Chapter II.
It says let $H$ be infinite dimensional, $V$ the set of ...
- 541
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$M^{\perp \perp}$ is the smallest closed subspace containing M
Let $H$ be a Hilbert space and $M$ an arbitrary subset. Show that $M^{\perp \perp}$ is the smallest closed subspace of $H$ that contains $M$.
I have checked this Smallest closed subspace which ...
- 664
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Is the isomorphism between $\mathcal{H} \otimes \mathcal{G}$ and the space of Hilbert-Schmidt operators from $\mathcal{H} \to \mathcal{G}$ bijective?
On page 35 of the review of Reproducing Kernel Hilbert Spaces by Muandet et al. (1), the authors write that
Is this isomorphism bijective? How can we show this? I know it is injective by the fact it ...
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Are two Hilbert-Schmidt operators pointwise equal iff the norm of their difference is zero?
Let $HS(H_1, H_2)$ be the Hilbert space of Hilbert-Schmidt operators from $H_1 \to H_2$. For $A, B \in HS(H_1, H_2)$ do we have pointwise equality $A = B$ (i.e. $Ax = Bx$ for all $x$) if and only if $\...
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Proof of Orthonormal basis for $L_2(\mathbb R)$ by Hermite Polynomials
Consider $L^2(\mathbb{R})$ as an Hilbert space with inner product $(\cdot ,\cdot)$. Define
$$\psi_n(x)=e^{-\frac{x^2}{2}}H_n(x),$$
where $H_n(x)$ is the Hermite Polynomials. How do you show that $\{\...
- 534
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What can be said about spectral convergence of compact operators?
Let $H$ be a Hilbert space and suppose that $T : H \rightarrow H$ is a self-adjoint compact operator. It has discrete spectrum of eigenvalues of finite multiplicity:
$$
\lambda_1 \geq \lambda_2 \geq ...
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Bounding the norm of the solution to a particular system of abstract linear equations
This is my very first question :) I have been working on this problem for the past couple of days, and I find myself really stuck.
The problem goes as follows: let $V$ and $W$ be real Hilbert spaces, $...
