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

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Hermitian matrices and great circles

I am considering parameterised curves in an $n$-dimensional complex vector space, given by the solution to the discrete Schrödinger equation: $$ |\psi\rangle(t) = e^{-iHt}|\psi_0\rangle, $$ Where $H$ ...
4
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1answer
392 views

Continuity of scalar product

In a Hilbert space $H$ with inner product and associated norm, why would if $\|x-x_n\| \longrightarrow 0$ and $\|y-y_n\| \longrightarrow 0$ also $\langle x_n,y_n\rangle \longrightarrow\langle ...
4
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2answers
219 views

a trace class operator problem

Could someone help me with this Prove that If $A$ and $B$ are positive trace class operators on a Hilbert space, then so is $A^zB^{(1-z)}$ for a complex number $z$ such that $0 <Re(z)< 1$. An ...
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1answer
175 views

Spectrum proofs

Let $T$ be a densely defined closed unbounded operator on a Hilbert space $H$. Show that if $\lambda$ is a point in the residual spectrum of $T$, then $\bar{\lambda}$ is in the point spectrum of the ...
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2answers
278 views

Trying to understand Hilbert Spaces…

I am trying to get a hold on Hilbert Spaces, but I am having difficulties combinging various definitions. I have looked it up on wikipedia and wolfram, there it states something like "A Hilbert ...
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665 views

Spectrum of an Orthogonal Projection Operator

I want to show that $ \sigma(p) = \{ 0,1 \} $ for any orthogonal projection operator $ p \notin \{ 0,I \} $ on a Hilbert space $ \mathcal{H} $. Recall that an orthogonal projection operator $ p $ on $ ...
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3answers
91 views

$\|\cdot\|_1 \leq \|\cdot\|_2 \leq \|\cdot\|_{\infty}$ for functions in $C([0,1])$?

Why does the following hold for continuous functions on $[0,1]$? $\|\cdot\|_1 \leq \|\cdot\|_2 \leq \|\cdot\|_{\infty}$
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262 views

Why is the numerical range of a self-adjoint operator an interval?

I was reviewing for a test for functional analysis when I came across the following statement: Let $T$ be a bounded self-adjoint operator on a Hilbert space $H$. Then the numerical range of it is ...
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1k views

$C[0,1]$ is not Hilbert space

Prove that the space $C[0,1]$ of continuous functions from $[0,1]$ to $\mathbb{R}$ with the inner product $ \langle f,g \rangle =\int_{0}^{1} f(t)g(t)dt \quad $ is not Hilbert space. I know that I ...
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1answer
322 views

Bounded operator from a Hilbert space to $\ell^1$ is compact

Let $H$ be any Hilbert space. How can we prove that any bounded linear operator $T\colon H \to \ell^1$ is compact? If we use the fact that the space $\ell^1$ has Schur property (norm and weak ...
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1answer
187 views

Proving that a sequence in $L^2(\mathbb R)$ is relatively compact

I have a bounded sequence $\{f_n\}_n$ in $L^2(\mathbb R)$ such that $\mbox{supp } f_n$ is uniformly bounded and $$ \int_{\mathbb R} x^2 |\Theta_n(x) (F f_n)(x)|^2 dx \leq C^2 $$ for all $n$, where ...
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1answer
93 views

Can Fourier transform be seen as a decomposition over a basis in a space of tempered distributions

Fourier series of a function that belongs to $L^2([0,T])$ can be seen as a decomposition of this function over an (orthonormal) basis in the Hilbert space $L^2([0,T])$. Fourier transform of a ...
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152 views

Is $\|x\| = \| \overline{x} \|$ in an inner product space?

Suppose $X$ is a complex inner product space of complex valued functions that is closed under conjugation. Is it true that $\|x\| = \| \overline{x} \|$ for all $x$? If not, is there a simple ...
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1answer
84 views

“Refinement” of the existence of faithful representations of C*-algebras?

Conway, in a course in operator theory, brings the statement 1. below as a theorem and statement 2. below as an exercise. Still, he states that 2. refines 1., but I can't see it. Every C*-algebra ...
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92 views

Upper bound for norm of Hilbert space operator

It is a standard result that for a bounded self-adjoint operator $T$ on a complex Hilbert space $H$, we have $||T||=\sup_{||x||=1}|\langle Tx,x\rangle|:=M$. It seems that for any bounded operator on ...
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98 views

(From Lang $SL_2$) Orthonormal bases for $L^2 (X \times Y)$

Lang $SL_2$ p. 13 :Let $\{\phi_i\}$, $\{\psi_i\}$ be orthonormal bases for $L^2(X)$ and $L^2(Y)$ respectively. Let $$\theta_{ij}(x,y) = \phi_i(x)\psi_i(y).$$ Then $\{\theta_{ij}\}$ is an ...
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1answer
65 views

How can we pick $f \in C(0,T;H)$ with $f(T) =0$ and $f(0) = h$, where $h$ is arbitrary?

Let $C(0,T;H)$ be the space of continuous functions $f:[0,T]\to H$ where $H$ is Hilbert. For every $h \in H$, why is it possible to pick a function $f \in C(0,T;H)$ such that $f(0) = h$ and $f(T) = ...
4
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1answer
129 views

functional analytic interpretation of the (co)variation and the doob decompostion

I have a question concerning the covariation of two time-discrete stochastic processes. Let $(\mathcal{F}_i)_{i\in T}$ be a filtration. We call a time-discrete, real-valued, adapted process $X$ ...
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2answers
185 views

Is duality an exact functor on Banach spaces or Hilbert spaces?

Let $V,V',V''$ and $W$ be vector spaces over $k$. Then, it is known that $\operatorname{Hom}(\cdot,V)$ is a contravariant exact functor, i.e. for each exact sequence $0\to V'\to V\to V'' \to 0$, and ...
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1k views

What is the difference between a complete orthonormal set and an orthonormal basis in a Hilbert space

I don't know if it's true but I think in a finite dimensional veccter space, an orthonormal set which is complete becomes a basis. But in a Hilbert space, the books say that the set of finite linear ...
4
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1answer
230 views

Spectrum in an separable Hilbert space

Let $H$ be a separable Hilbert space with orthonormal basis $\{e_i\}$. Let $(c_n)$ be a bounded sequence of complex numbers and consider the bounded linear operator $T$ on $H$ defined by $$Tx = ...
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1answer
929 views

How to prove this integral operator is compact?

$T_kf=\int K(x,y)f(y)dy$ where $K(x,y)=\frac{\phi(x)\phi(y)}{|x-y|^{n-\alpha}}$ $\phi(x)$ is a smooth function on a compact support. $f$ is defined on $R^n$ and $K$ is defined on $R^n\times R^n$ ...
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329 views

Show that if the Riesz map is surjective on $H$, then $H$ is a Hilbert space

Let $H$ be a vector space equipped with an inner product $(\cdot, \cdot)$ and $f:H\to H',\ f(x)=(\cdot,x)$ surjective. Now, why $H$ is a Hilbert space? The other direction is clear by Riesz' ...
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131 views

Is it possible to 'approximate' compact, convex sets in $\ell^2$ by the Hilbert cube

Define $H=\{(x_n)_n\in\ell^2:|x_n|\le \frac1n, n\in\mathbf N\}\subset\ell^2$. This set is known as the Hilbert cube and it is well-known that $H$ is compact, convex and non-empty. Let ...
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1answer
654 views

positive invertible operators

I need the following result. I think it's quite obvious but I don't know how to prove that: Let $C, T : \mathcal{H} \rightarrow \mathcal{H}$ be two positive, bounded, self-adjoint, invertible ...
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94 views

Weak convergence as convergence of matrix elements

Let $H$ be a Hilbert space with orthonormal basis $(e_h)_{h \in \mathbb{N}}$ and let $(A_n)_{n \in \mathbb{N}}$ and $A$ be bounded linear operators. We say that $A_n$ converges weakly to $A$ if ...
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1answer
398 views

Direct sum $\Rightarrow$ Direct Integral, Tensor product $\Rightarrow$?

Is there a way to define a tensor product over a measure space(=index set) with a continuous measure for Hilbert spaces? For the sum we have the notion of a direct integral, here.
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32 views

If the scalar product are equal then the operators are equal.

I want to show the following: Let H be a $\mathbb C$ -hilbert space and $S,T\in L(X)$ If $\langle Sx,x \rangle = \langle Tx,x \rangle$ for all $x\in H$, then $S=T$ Any hints for me?
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86 views

Uncountable series without axiom of choice

Consider a sequence of positive real numbers $(\alpha_i)_{i\in I}$ for some (suppose maybe wellordered for now) set $I$. Using axiom of choice, it is easy to see that $\sum_i \alpha_i$ is infinite if ...
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2answers
142 views

Is this following bilinear form coercive?

First of all I want to mention that this is homework, so don't spoil it and reveal all the answer. just some guidenss :) Let $H$ be a Hilbert space, $T:H\rightarrow H$ a bounded linear operator for ...
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291 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 refernce to this paper [Olivier Chapelle, ...
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55 views

$\left\{x\in H: 2\leq \|x\|\leq 5\right\}$ is compact?

In a Hilbert space $H$ of dimention infinite, $A=\left\{x\in H:2\leq \|x\|\leq 5\right\}$ is compact? (totally bounded and complete) Thanks in advance.
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1answer
52 views

Finding an orthonormal basis from an existing one in a Hilbert space

Suppose we are given a separable Hilbert space $H$ with countable orthonormal basis $\{e_n\}$. Suppose we are given an orthonormal set $\{f_n\}$ such that $\sum\|e_n-f_n\| < 1$. How do we prove ...
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361 views

When do inner products of weakly convergent subsequences converge?

If we have 2 weakly convergent subsequences in $L^2(U)$ (for $U$ some bounded open domain with smooth boundary), $u_k\rightharpoonup u$ and $v_k\rightharpoonup v$, under which conditions do we have ...
4
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144 views

find a weak solution in an intersection of Sobolev spaces

In using lax-milgram to find a weak solution in an intersection of sobolev spaces the weak solution for $$ -\Delta^2 u = f \in L^2(U)\\ \\ u|_{\partial U}=\Delta u|_{\partial U} = 0 $$ was ...
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1answer
122 views

Question about adjoint map and strong operator topology (SOT)

I am wondering if there is any condition one can apply (e.g. uniform boundedness?) that ensure the adjoint of a net of SOT-continuous elements is again SOT-continuous? My major question is ...
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209 views

about weak convergence in $L^{2}(0,T;H)$

I am trying to do an exercise and if the affirmation below is true, my exercise is done . This is the affirmation : Affirmation : Let $H$ a Hilbert space and suppose $u_k$ converges weakly to $u$ ...
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60 views

Selfadjoint and continuous operator on a complex Hilbert space

Let $T\colon H\to H$ be a selfadjoint continuous operator on a complex Hilbert space. Show: $$ \lVert (T\pm i\mbox{Id})x\rVert^2=\lVert Tx\rVert^2+\lVert x\rVert^2~\forall~x\in H. $$ -- How can I ...
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1answer
761 views

How to find an orthonormal basis for $L^2(\mathbb{R},\mathbb{C})$?

Consider the Hilbert space $X:=L^2(\mathbb{R},\mathbb{C})$ Now consider the operator that takes the second derivative, i.e. $A := \partial_{x}^2$, i.e. $A: H^2(\mathbb{R},\mathbb{C}) ...
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178 views

Orthogonality checking in Kreyszig exercise

Let $H$ be inner product space with inner product $\langle\cdot,\cdot\rangle$ and norm $\lVert \cdot\rVert$. Let $x,y \in H$. Would you help me to prove that $\langle x,y\rangle=0$ if and only if ...
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83 views

Invertible operator not preserving Hilbert dimension

It is known that for a bijective linear operator $T:X\to Y$ the algebraic dimensions of the linear spaces $X$ and $Y$ coincide. I am asking for an example of an invertible (bounded) linear operator ...
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423 views

Dense subset of a Hilbert space

Let $A$ be a dense subspace of a Hilbert space $H$. Denote $\ell^2$ the Hilbert space of (complex valued) square-summable sequences and denote $\ell^2(H)$ the Hilbert space of $H$-valued ...
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136 views

The union of cyclic subspace is also a cyclic space

Given a separable Hilbert space $H$, $U$ is a unitary operator. A cyclic subspace, denoted as $Z(x)$ for some $x\in H$, is defined as the closure of linear span of $U^nx$, where $n\in \Bbb Z$ is any ...
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1answer
295 views

Set of all compact operators $K(H)$ is the unique ideal in $B(H)$?

I want to show that the set of all compact operators $K(H)$ is the unique ideal in $B(H)$. Is there any relation between invertibility and compactness of an operator?
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463 views

Convergence of a sequence of periodic functions

Motivated by the homogenization theory which studies the effects of high-frequency oscillations in the coefficients upon solutions of PDE, I am thinking about the following question. Let the ...
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195 views

Tensor Product: Hilbert Spaces

This question has been modified... Problem Given Hilbert spaces. In general, their algebraic tensor product isn't complete: ...
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1answer
152 views

For a Hilbert space $\mathcal{H}$, is every bounded linear operator on $\mathcal{H}$ a linear combination of unitary operators?

Let $(\mathcal{H}, (\cdot, \cdot))$ be a Hilbert space, and let $B \in \mathcal{B}(H)$ be a bounded linear operator on $H$. If $\mathcal{H}$ is a complex Hilbert space, then $B$ can be written as a ...
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1answer
206 views

Reproducing Kernel Hilbert Space (RKHS) constructed by the summation of positive-definite kernels.

Let $k_1,\ldots,k_p$ be positive definite kernels defined on $\mathcal{X}\times\mathcal{X}$, where $\mathcal{X}$ is a non-empty set. $k_i$ is the reproducing kernel of the Reproducing Kernel Hilbert ...
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2answers
44 views

Orthogonal representation of finite operator

I would like to know if my proof is correct. Statement: Let $T$ be a finite rank operator on a Hilbert space $\mathscr{H}$. Show that $\forall \, h \, \in \mathscr{H}, \, T(h)$ can be written as ...
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1answer
114 views

Spectral decomposition of normal operator

Define $T$ from $L_{2}(R)$ into itself by $T(f)(t)=f(t+1)$. Show that $T$ is normal and finds its spectral decomposition. I've shown that $f$ is normal (in fact it's unitary) but how do I find its ...