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Questions tagged [hilbert-spaces]

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

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A Hilbert space is separable if and only if it admits a countable orthonormal basis [duplicate]

According to Wikipedia: A Hilbert space is separable if and only if it admits a countable orthonormal basis While this statement seems very reasonable, it is not clear to me how one would go about ...
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42 views

Polynomial ring as a (complete) metric space

Let $k$ be a field of characteristic zero (for example $\mathbb{R}$ or $\mathbb{C}$), and let $R=k[x_1,\ldots,x_n]$ be the $k$-algebra of polynomials in $n$ variables $x_1,\ldots,x_n$, $n \geq 1$. ...
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Isomorphisms and Implicit Function Theorem in the context of Hilbert spaces

Let X be a Hilbert space with inner product denoted $(\cdot,\cdot)$ and norm $\|\cdot\|$ induced by it. Suppose $F\,\colon\,X \times \mathbb{R} \to X$ is continuously Frechet differentiable with ...
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1answer
22 views

A condition to ensure that adjoint and inverse commute

Let $A$ be a densely defined operator on a Hilbert space $H$. Let us assume that $A$ is injective and $Ran(A)$ is dense. Then I want to show that $$ (A^*)^{-1} = (A^{-1})^*. $$ So far, I was able to ...
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1answer
37 views

A question about self-adjoint operators

Let $A$ be a densely defined symmetric operator on some Hilbert space $\mathcal{H}$. Let's say that we know that there exists $(A + 1)^{-1}$ and that this is a self-adjoint operator. How can I prove ...
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46 views

Show that finitely generated cone in Hilbert space is closed.

Let $X$ be a real Hilbert space. Let $\{c_i\}_{i=1}^m\subset X$. Denote $$ Y=\left\{\sum\limits_{i=1}^m\lambda_ic_i\Big|\lambda_i\in[0,+\infty),\quad 1\leq i\leq m\right\}. $$ Show that $Y$ is a ...
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4answers
52 views

Hilbert space and orthonormal basis.

Let $H$ be a Hilbert space and let ${e_n} ,\ n=1,2,3,\ldots$ be an orthonormal basis of $H$. Suppose $T$ is a bounded linear oprator on $H$. Then which of the following can not be true? $$(a)\quad T(...
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Lagrange multiplier method on Banach spaces (in Dirac notation)

I want to prove Cauchy–Schwarz' inequality, in Dirac notation, $\left<\psi\middle|\psi\right> \left<\phi\middle|\phi\right> \geq \left|\left<\psi\middle|\phi\right>\right|^2$, using ...
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3answers
52 views

Prove that a subset in Hilbert space is open

I'm a undergraduate who enjoys math and follows his first course on real Analysis. I follow this course for fun and i'm stuck on the following problem: Let $H^{\infty}$ be the Hilbert cube (the ...
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45 views

Reference Request: KKT in Hilbert Space

Are there analogues of Slater's condition and the KKT conditions in separable Hilbert spaces? Does the infinite dimensionality pose a problem?
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Show Cauchy-Schwarz inequality from triangle inequality for $\mathbb{C}$-vector space

In many textbooks the proof of Cauchy-Schwarz inequality needs to introduce a parameter $\lambda$ to take $\langle x+\lambda y,x+\lambda y \rangle$ as the first step. I am trying to proof the Cauchy-...
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GNS representation and states of $C^*$-Algebras

Let $\varphi$ be a state of the $C^*$-algebra $A$, $B\subset A$ a hereditary subalgebra and $K_\varphi:=\{x\in B : 0\le x \le 1, \varphi(x)=1\}$. Let $\pi_\varphi:A\rightarrow \mathcal{B}(H_\varphi)$ ...
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2answers
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$T$ is an isometry if and only if $\langle Tx, Ty \rangle = \langle x, y \rangle$

I want to prove: A linear mapping $T:X \to Y$ between two pre-Hilbert spaces is an isometry if and only if the inner products $\langle Tx, Ty \rangle = \langle x, y \rangle$ for all $x, y \in ...
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1answer
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Spectrum included in spectral measure support

Let $\mathcal{H}$ be an Hilbert space and $A=\displaystyle{\int} \lambda \,dE(\lambda)$ a spectral operator on $\mathcal{H}$ associated to the spectral measure $E$. I am interested in the very basic ...
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2answers
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A linear bounded operator has to be continuous?

I just found on the wiki https://en.wikipedia.org/wiki/Continuous_linear_operator stating that An operator between two normed spaces is a bounded linear operator if and only if it is a continuous ...
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Show the average of first n terms of a power series converges in Hilbert space

Let $X$ be a Hilbert space. Let $A:X\mapsto X~$ be a continuous linear operator such that $||A||=1$. For any $x\in X$, define a sequence $\{y_n\}_{n=1}^{\infty}$ as \begin{equation*} y_n=\frac{1}{n}\...
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Why a countable set of numbers can characterize a function in Hilbert space?

I'm a new graduate student major in physics; the only mathematical course I was formally trained is Real Analysis, so my following statement of my question may seem a little unprofessional, and I hope ...
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Image of a dense set [closed]

Is the image of a dense set under an isometric operator is again dense set? i.e., Given two Hilbert spaces $X, Y$. If $T:X\to Y$ is an isometric operator and $S$ is a dense span subset of $X$, is ...
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1answer
41 views

Hilbert-Schmidt operator defined by non-orthogonal basis

I have the following operator on $L^2(0,1)$ $$ Tf = \sum_{n \geq 0} 2^{-n}\langle f,v_n\rangle v_n$$ Where $v_n(t) = t^n$. I was able to prove that it is Hilbert-Schmidt, but now I need to ...
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Proving a proposition about special functions in finite dimensional Hilbert space

We call frame function any mapping $f : \{ x \in H, \| x \|=1 \} =:\mathbb{S}(H) \mapsto \mathbb{R} \cup \{ \pm \infty \}$ (here $H$ denotes a finite dimensional Hilbert space) satisfying the ...
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1answer
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Projection onto the span of linearly independent vectors in Hilbert spaces

Let $H$ be a Hilbert space and $S\subseteq H$ a closed subspace. Moreover, let $\{s_{n}\}_{n=1}^{\infty}\subseteq S$ a complete and linear independent sequence in $S$, i.e. $S=\overline{\text{Span}\...
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1answer
38 views

Find orthogonal basis for $V^{\perp}$ and find orthogonal projection on $V$.

I have a task in the functional analysis exam that I could not do. Problem: In Hilbert space $L_2 [0,1]$ (with Lebesgue measure) let: $$V = \left\{ f \in L_2 [0,1]: \int_{0}^{1} t \cdot \overline{f(...
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Measurable function on Hilbert space

Every continuous function defined on $\mathbb{R}$ is Borel measurable. Is it also true in Hilbert spaces ? any references are there?
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The Hardy space $\mathbb{H}^{n\times m}_{2}$ in the upper half plane

Given $f\in \mathbb{H}^{1\times m}_{2}(U) $ and $g\in\mathbb{H}^{m\times m}_{2}(U)$. Is it true that $fg\in \mathbb{H}^{1\times m}_{2}(U)$ ?? If not, what conditions needed so that it would be ...
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How can we have a complex basis for a real-valued space?

I have started studied functional analysis few times ago and now I am stuck in a doubt which I think is really trivial, but I cannot still find an answer. Consider the Hilbert Space $X = L^2[-\pi,\pi]...
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2answers
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Showing that the space $L^2(\Omega \rightarrow H)$ of Square-integrable Hilbert-valued functions is a Banach Space

I am trying to fill in the details of Nate Eldredge's answer to "Is this set of random variables a Hilbert space?". In particular, let $(\Omega,\mathcal{F}, \mu)$ be a measure space and $H$ be a ...
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Is there a Hilbert space containing exactly 128 elements? [closed]

We can construct a Metric Space having exactly 128 elements. In fact we can define a discrete metric space on finite set containing 128 elements. The questions are "Can we construct a Vector Space/...
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1answer
50 views

A question about a theorem in *Quantum dynamical semigroups generated by noncommutative unbounded elliptic operators*

I'm studying the paper Quantum dynamical semigroups generated by noncommutative unbounded elliptic operators by Changsoo Bahn and Chul Ki Ko and Yong Moon Park(https://arxiv.org/abs/math-ph/0505026v1)....
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1answer
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Canonical isomorphism between finite-dimensional Hilbert spaces

I want to ask is there a canonical isomorphism between two same finite dimensionality Hilbert spaces. My intuition is telling me no, there is an isomorphism, but not canonical. The reason is because ...
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3answers
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Orthonormal basis in every Hilbert space?

Is it true that in every Hilbert space (not separable) there exists an orthonormal basis (complete and countable orthogonal system)? So equivalently asked can we have a Hilbert space in which it is ...
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1answer
51 views

Non-orthogonal extension of an inner product

Let $W$ be a real linear vector space of possibly infinite dimension and let $<\cdot, \cdot>_W$ be an inner product on $W$. Further let $V$ be a superspace of $W$, such that holds $V = W \...
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Is the simplicial complex with vertices the critical points of a $C^2$-function on a Hilbert manifold $X$, homotopy equivalent to $X$?

Let $X\subset\mathbb{R}^{\infty}$ be a paracompact connected Hilbert manifold. Moreover, let $f:X\to\mathbb{R}$ be a $C^2$-function. Then is the simplicial complex $\Delta:=\{p\in X:df(p)=0\}$, ...
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How do we compute homology from the Čech complex $\text{Čech}(r)$ with $B^{\infty}$ balls covering the Hilbert manifold $M$?

Suppose a Hilbert manifold $M$ is covered by the union of $\infty$-balls (in the sense of Baire spaces), namely $M=\bigcup_{\alpha\in A}B^{\infty}$, without knowledge of intersections. The only ...
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2answers
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Convergent sequence in Hilbert space

Let $H$ be a Hilbert space with orthonormal basis $(e_n)_n$ and $(a_n)_n \in \ell^2(\mathbb{N})$, so $$\sum_{n=0}^{\infty} |a_n|^2 < \infty$$ Now, define the sequence $(\xi_n)_n$ in $H$ as $$\xi_n= ...
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The average effect of a self-adoint projection

First, here is a brief non-rigorous version of the question: Staring with two arbitrary rays (one-dimensional subspaces) in a finite-dimensional Hilbert space, what is the average effect of a ...
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$\lim_{n \to \infty} \langle\bar A_nx,y\rangle = \langle Ax,y\rangle$

Let $A$ be an operator on a separable Hilbert Space $H$. So we can consider $A=(a_{ij})_{1\leq i,j\leq \infty}$. Let $\tilde A_n = $$ \begin{pmatrix} A_n & 0 \\ 0 & -I \end{...
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2answers
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An application of Riesz representation theorem $\lim_{n\rightarrow \infty} \left< x_n,e_k\right>=\left< x,e_k \right>$.

Suppose that $\mathcal{H}$ be a real, separable Hilbert space and $\{e_k \}$ be a countable orthonormal basis. And let $x\in \mathcal{H}$ and $\{x_n\}\subset \mathcal{H}$ is a bounded sequence. Than ...
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Semi-inner product structure in complex Hilbert spaces

Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $(F,\langle\cdot,\cdot\rangle)$. Any $M\in \mathcal{B}(F)^+$ (i.e. $\langle Mx\;, \;x\rangle\geq 0$ for all $...
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1answer
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One-Dimensional Subsets of Hilbert Space vs. Euclidean Space

Motivated by a question I received (though a bit different), is there a one-dimensional subset of Hilbert Space that cannot be embedded in $\mathbb{R}^3$? Here, dimension refers to covering or ...
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1answer
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If f is bounded ($\vert f(x)\lvert \leq M$), and $f\in L^1(R)$, then

I’m doing this problem: If f is bounded ($\vert f(x)\lvert \leq M$), and $f\in L^1(R)$, then: $f\in L^2(R)$ with $\lvert\lvert f\lvert\lvert_{L^2(R)}\leq M^{1/2}\lvert\lvert f\lvert\lvert_{L^1(R)}^{1/...
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Is the Banach space of continuous functions $I \to \ell^2$ separable?

Inspired by this question. Consider the vector space $V$ of all continuous functions $I \to \ell^2$ for $I=[0,1]$ the closed unit interval and $\ell^2$ the Hilbert space of all square-summable ...
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Why do we need to use dominated convergence theorem?

I was thinking of this problem: If $f_n\rightarrow f$ pointwise a.e. and $\lvert f_n\lvert \leq g$ for some $g\in L^p$, then prove that $f_n \rightarrow f$ in $L^p$. To use the dominated convergence ...
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Proof explanation related to the operator matrices

Let $F$ be a complex Hilbert space. Let $A,B,C,D\in \mathcal{B}(F)$. Consider the operator matrix $T$ such that \begin{equation*} T=\begin{pmatrix}A & B \\ C & D \end{pmatrix}\in \mathcal{B}(F\...
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1answer
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Suppose $f_n\rightarrow f$ Point wise a.e.

Suppose $f_n\rightarrow f$ pointwise a.e. Show that $f_n\rightarrow f$ in $L^1$ if and only if $\lvert\lvert f_n\lvert\lvert_1 \rightarrow \lvert \lvert f\lvert\lvert_1$. What happens if it’s all in ...
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1answer
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Let $f\in L^p (R)$ for $1\leq p<\infty$ and let $f_t(x)=f(x-t)$.

Let $f\in L^p (R)$ for $1\leq p<\infty$ and let $f_t(x)=f(x-t)$. Prove each $f_t\in L^p(R)$. Prove $\lvert \lvert f_t-f\lvert\lvert_p\rightarrow0$ as $t\rightarrow0$. What happens as $t\rightarrow ...
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1answer
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Suppose $\mu$ and $\nu$ are finite positive measures on a measuable space

The problem is as follow: Suppose $\mu$ and $\nu$ are finite positive measures on a measuable space (X, M). Show there is $f\in L^1(X,\mu)$ so that $\int f \ d\mu=\int (1-f) \ d\nu$. I know that $f\...
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1answer
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For an orthonormal set in $L^{2}[0,1]$, show that $ \int_{A} \sum_{k=1}^{\infty} |f_{k}(x)|^{2} \geq 1$ for a non-zero measurable subset $A$.

I was stuck on the following question If ${f_1, f_2, . . . }$ is a complete orthonormal set in $L^2 [0, 1]$ and $A$ is an arbitrary subset of positive Lebesgue measure in [$0,1]$ show that $$ \...
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1answer
58 views

Is there an orthonormal basis for $L^2([0,1])$ consisting of only even-degree polynomials?

Problem: Does $L^2([0,1])$ equipped with the inner product $$\langle f,g \rangle = \int_0^1 \overline{f(x)}g(x) \, dx$$ have an orthonormal basis of the form $\left\{f_n(x):=\sum\limits_{k=0}^n a_k ...
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60 views

Dirac delta function like a “function” of an operator

Let $\mathcal{S}\subset \mathcal{H}=L^2(\mathbb{R}^n)\subset \mathcal{S}^*$ be the Schwartz space, the Hilbert space and the space of tempered distributions respectively. Consider an algebra $\mathcal{...
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Composition of absolutely continuous function

if I have a function $x:[0,\infty)\rightarrow H$ (H is an hilbert space) that is absolutely continuous (AC), why should the function $\|x(t)-y\|^2$ be AC, for $y\in H$? Yet again, why should $\|x_{\...