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Questions tagged [hilbert-spaces]

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

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BLP in terms of a Hilbert space

I am reading Stein's 2013 monograph 'Interpolation of Spatial Data' where he characterizes the best linear predictor (BLP) in terms of a Hilbert space. Let $Q$ be the set on which a random process $Z$...
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Uniform convergence of polynomial approximation on Schwartz space

I have a question regarding uniform convergence of basis expansion in Schwartz space. For $L^2(\mathbb{R},\lambda)$, $\lambda$ Lebesgue measure, the partial sums of basis expansion (Hermite functions) ...
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$\mathcal{F}$ convex and lower continous $\Rightarrow$ $\mathcal{F}$ weakly lower continous

I'm having troubles with one part of a problem consisting out of several subquestions and hope some of you can help me! Let $X$ be a Banach-space and let $\mathcal{F} : X \rightarrow (-\infty,\...
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Corollary of Projection onto a closed convex set and geometric interpretation

I need help with geometric interpretation of this theorem and with the corollary of the theorem: Theorem: projection onto a closed convex set Let $K \subset H$ be a nonempty closet convex set. ...
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Non-expansiveness implies strict pseudo‑contraction in a Hilbert space

I know that non-expansiveness implies pseudo‑contraction while strict‑contraction implies strict pseudo‑contraction. However, the authors of this paper (page 4) stated that strict pseudo‑contraction ...
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Equivalence of statements about bounded linear maps on a Hilbert space

Assume $H$ is a Hilbert space and $V \in L_b(H)$. I want to show, that the following propositions are equivalent: V is an isometry. For every orthonormal system $\{u_{\alpha}: \alpha \in A \}$ the ...
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Right Hilbert space for a duality pairing

Let $B$ be a Banach space. Let $B^*$ be its dual, and let $K\subseteq B^*$ be some linear dense subspace. Denote the duality pairing between $B$ and $B^*$ via $\langle\cdot ,\cdot \rangle$. Suppose we ...
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Every surjective isometry on a Hilbert space is indeed a unitary operator

I have a little bit confused on unitary operators and surjective isometries on a Hilbert space. I think it is quite clear that A operator is unitary if and only if it is a surjective isometry. ...
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Expectation of an inner product in an infinite dimensional Hilbert space

Let $\mathcal{H}$ be a Hilbert space with the Borel $\sigma$-algebra. Let $(\Omega, \mathcal{F}, P)$ be a probability space and $x,y$ two $\mathcal{H}$-valued random variables, i.e. measurable maps ...
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1answer
27 views

Momentum operator in the position basis

J.J Sakurai shows in the section of ' Momentum operator in the position basis' as $P$$\lvert\alpha\rangle$=$\int dx^{'}\lvert\ x{'}\rangle\Bigl(-i{h\over 2\pi}$ $\partial\over\partial x{'}$$ \...
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Decomposition of an unbounded operator

Suppose $H=H_1\oplus H_2\oplus...$ is an orthogonal decomposition of a Hilbert space $H$. Let $T:\mathrm{dom}(T)\subseteq H\to H$ a densely defined linear operator (it could be not bounded). For each $...
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Construction of an operator $A$ on $\ell^2(\mathbb{N}^*)$ satisfying a property

Let $\ell^2(\mathbb{N}^*)$ be the Hilbert space with the inner product $$\langle x\mid y\rangle_2:=\sum_{i=1}^{+\infty}x_i\overline{y_i},\;\,\forall\,x, y \in \ell^2(\mathbb{N}^*).$$ Consider the ...
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What is $\ell^2(\Gamma)$ for a discrete group $\Gamma$?

I am trying to get my head around the left regular representation of a group, and I am not sure of the definition of the space $\ell^2(\Gamma)$ if $\Gamma$ is a discrete group. To quote what I am ...
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What are the eigenstates of $X^N$ operator?

NOTE: I have first asked this on physics.stackexchange, they advised me to ask on math.stackexchange The operator $X$ is called the position operator in physics with it's conjugate being the momentum ...
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Can this exponential be complex valued?

My complex analysis is very sketchy, and I am a little stumped by the following - although it seems incredibly innocuous. For $t\in\mathbb R$ and a fixed parameter $\alpha\in\mathbb R/\{0\}$ does it ...
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Hilbert Spaces and linear subspaces

Let X and Y be linear subspaces of a Hilbert space $\mathcal{H}$. $\\$ Recall that $\\$ $X + Y$ = {$x+y: x \in X, y \in Y$} $\\$ Show that $\\$ $(X+Y)^\bot$ = $X^\bot\cap Y^\bot$ $\\$ I tried ...
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Projection estimator in hilbert space [closed]

Let $ f \in \mathbb{L}^2[0,1] $ $ Y_i = f(i/n) + \epsilon_i $ with $ \epsilon$ independent and centered With basis $ \phi_1 (x) = 1 , \: \: \phi_{2k} (x) = \sqrt{2}cos(2\pi kx) \: \: \phi_{2k+1} (...
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Is there a norm making $C([0,1])$ into a Hilbert space?

The space $C([0,1])$ of continuous functions on $[0,1]$ is an inner product space under the $L^2$-norm, but not complete. Equipped instead with the $L^\infty$-norm, it becomes complete but the norm is ...
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$L^2$ representation of Hilbert subspace of $L^2$

Suppose that $\mathcal{H}$ is a Hilbert space which is a subspace of $L^2(X,\mathcal{A},\mu)$. Does there exist a sub-$\sigma$-algebra $\mathcal{B}$ of $\mathcal{A}$ such that $\mathcal{H}=L^2(X,\...
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Rewriting elements of the closure of a countable set as an infinite sum of the spanning element in Hilbert spaces.

Let $H$ be a Hilbert space and $f \in \overline{span\{\phi_{m} : m \in \mathbb{Z}\}}$ with $\phi_m \in H$ for every $m \in \mathbb{Z}$ . Why can we write \begin{equation} f = \sum_{m \in \mathbb{Z}} ...
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The vanishing property of a compact operator acting on an orthonormal basis of a Hilbert space

The question is: Suppose $H$ is a Hilbert space, $A \in K(H)$, and $\{e_n\}$ is an orthonormal basis of $H$. Prove $\lim\limits_{n \to \infty} \langle Ae_n, e_n \rangle = 0$. $\langle \cdot, \cdot ...
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29 views

Product of two Linear Operators

Let $S(e_n)=e_{n+1}$ and $T(e_n)=e_{n+2}$ be two linear operators on the Hilbert space $l_2(N)$, the space of all sequences $\sum_{1}^\infty |a_k|^2 < \infty$, and $\{e_n\}, n=0,1,2,...$ is the ...
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Are these two norms on the dual space of a Hilbert space equivalent?

Let $\mathcal{H}$ be a Hilbert space, and $\mathcal{H}^*$ its topological dual space (the space of continuous linear forms on $\mathcal{H}$). The exists a conjugate-linear isometry between these two ...
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Convergence of trace class operators in Hilbert Schmidt norm

Let $\mathscr{A}_n$ be a sequence of trace-class operators on a Hilbert space $\mathcal{H}$ and let further $\mathscr{A}$ be another trace-class operator on the same space. Assume that $\mathscr{A}_n$...
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General Formula of a Linear Operator given its act on the Standard Orthonormal Basis

I am trying to find a general formula for a linear operator on a Hilbert space when its action on the standard orthonormal basis is known. I include my work below. Please tell me whether my solution ...
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Finding Adjoint Linear Operators in a Hilbert Space

I am looking for verification of my attempt in finding the adjoint operator of a linear operator. Let $S(e_n)=e_{2n+1}$ be a linear operator in the Hilbert space $l^2(N)$, the space of all summable ...
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1answer
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Unique decomposition of a vector in finite-dim Hilbert space

Let $\mathcal{H}$ be a finite-dimensional Hilbert space and $L$ and $L^{\perp}$ be a subspace and its orthogonal complement such that $$L\oplus L^{\perp}=\mathcal{H}$$. Show that any vector $\...
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Why the space have to be complex in the spectral theorem for bounded self-adjoint operators?

In the spectral theorem for compact self-adjoint linear operators $T:H\to H$ (as stated in Conway's book), the Hilbert space $H$ can be real or complex. However, in the spectral theorem for bounded ...
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Must a subspace of a Euclidean space with zero orthogonal complement be dense?

Let $X$ be a Euclidean space (i.e. a vector space over $\mathbb{R}$ or $\mathbb{C}$ with an inner product, $\textbf{not necessarily complete}$). Let $S$ be a subspace of $X$ with its orthogonal ...
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understanding Hilbert Sobolev spaces

We denote the space $\dot{H}^k$ for the homogeneous Sobolev space and $H^k$ for the inhomogeneous Sobolev space where $H^k=W^{k,2}$. It is true that de dual of $\dot{H}^k$ can be identified with $H^k$...
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Hardy inner product for the right half plane

The Hardy space $H^\infty(\mathbb{C_+})$ is defined to be the space of all functions $f$ >analytic and bounded on the complex right half plane $\mathbb{C_+}$ with the norm $\lVert \cdot \rVert_H$ ...
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Are 2 Hilbertspaces with different inner products identical if the associated norms are equivalent?

suppose we have the Sobolev space $H^1_0(\Omega)$ over a bounded domain $\Omega \subset \mathbb{R}^2$. With the standard inner product it sure is a Hilbert space. BUT: What if we equip $H^1_0$ with ...
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Showing that $0\leq A\leq B$ and $B \in \mathcal{L}_c(H)$ implies that $A \in \mathcal{L}_c(H)$.

Exercise : Let $H$ be a Hilbert space and $A,B \in \mathcal{L}(H)$ be self-adjoint operators with $0 \leq A \leq B$ and $B \in \mathcal{L}_c(H)$. Show that $A \in \mathcal{L}_c(H)$. Thoughts : ...
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Square root of positive matrix operator

Let $F_1,F_2$ be two complex Hilbert spaces. Consider \begin{equation*} T=\begin{pmatrix}A & B \\ C & D \end{pmatrix}\in \mathcal{B}(F_1\oplus F_2). \end{equation*} If $T$ is a positive ...
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Determine the matrix representation for an operator written as an outer product

Suppose $|v_{i} \rangle$ is an orthonormal basis for an inner product space $V$. What is the matrix representation for the operator $|v_{j}\rangle \langle v_{k}|$, with respect to the $|v_{i}\rangle$ ...
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An isomorphism between the dual space of a Hilbert space and a larger space containing the same Hilbert space

Let $G \subset H \subset F$ be three Hilbert spaces such that the smaller ones are continuously and densely embedded into the larger ones. Furthermore, assume $$|\langle h, g\rangle_H|\le \|h\|_{F}\|g\...
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Distributivity of tensor product over a direct sum

Let $\mathcal{H}, \mathcal{K}$ be finite dimensional Hilbert spaces and consider the space $$\left(\mathcal{H} \oplus \mathcal{K}\right) \otimes L^2(\mathbb{R}).$$ I would like a reference to show ...
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Showing that an orthonormal set becomes a basis for the Hilbert space

This is an exercise from Folland Real Analysis Chapter 8 that I am stuck at. I am actually stuck at (b). I succeeded in showing that $H_a$ is a Hilbert space and the given set is an orthonormal set of ...
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$L^2$ functions with compactly supported Fourier transforms form a Hilbert space

Given a fixed compact subset of $\mathbb{R}$, I want to show that square integrable functions on the real line whose fourier transforms are supported in the given compact set form a Hilbert space in ...
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Proving Lipschitz Continuity

Let $X$ be a Hilbert Space, $\varphi \in C^{2}(X,\mathbb{R})$, $c\in\mathbb{R}$, and $\varepsilon > 0$. Denote $\varphi^{-1}(\,\cdot\,)$ as the pre-image of $\varphi$ and define \begin{align*} A &...
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Showing that the bounded $C \subseteq L^p[0,1]$ is uniformly integrable.

Exrcise : Let $C \subseteq L^p[0,1], 1 < p < \infty$ be bounded. Show that $C$ is uniformly integrable. Attempt : It is $L^p[0,1] \subseteq L^1[0,1]$ and $L^p[0,1] \hookrightarrow L^1[0,1] \...
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When a seminormed space is complete?

Let $(F, \langle \cdot\;,\;\cdot\rangle)$ be a complex Hilbert space. Let $M$ a positive semidefine operator on $F$. Consider the following positive semidefinite sesquilinear form: \begin{eqnarray*} ...
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Spectral Theorem: Realization of a direct sum of $L^2$ spaces as a single $L^2$ space

The following is motivated by an attempt to understand the Spectral Theorem for Bounded operators on a none separable Hilbert space. One version of the theorem states that for a bounded (say normal) ...
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Approximation of subset of Hilbert space by finite-dimensional functions

I cannot come up with an answer to the following problem, which I came across: Let $H$ be a separable, real Hilbert space with ONB $\{e_n\}_{n \in \mathbb{N}}$ and let $U \subseteq H$ be open (if it ...
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Dense subspaces of a tensor product

Let $\mathcal{H}$ be a Hilbert space (infinite dimensional, in general) then consider its tensor square: $\mathcal{H} \otimes \mathcal{H}$. Is the space $$\{f\otimes f \ | \ f\in \mathcal{H}\}$$ Dense ...
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Intersection of dense subspace with an affine subspace is dense in the original space intersected with said affine subspace?

Let $\mathcal{H}$ be an infinite dimensional Hilbert space equipped with a norm $\|\cdot\|$. Let $U\subseteq \mathcal{H}$ be a subspace of $\mathcal{H}$ which is dense in the norm $\|\cdot\|$, in the ...
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Is every linear finite element space over a bounded domain a subspace of the sobolev space H^1?

Since my knowledge of functional analysis, $L^p$-, Sobolev- and Hilbert spaces is not very good, I thought I could ask... Suppose we have a domain $\Omega \subset \mathbb{R}^2$ which is continuously ...
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$L^1$ and $L^2$ spaces, how to determine which to work with?

This is from a standing point of a new student of measure-theoretic probability. For example, we have the following two definitions of conditional expectation: Definition 1 ($L^2$): Let $(\...
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Every basis in a finite dimensional Hilbert space is a Riesz Basis

The goal is to prove that every basis in a finite-dimensional Hilbert space is a Riesz basis, i.e., there exist constants $A>0$ and $B>0$ for the basis $\{x_k\}$ such that: $$ A \sum_n |a[n]|^2 \...