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

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45 views

Positive operators in Hilbert spaces

Let $H$ be a Hilbert space. I am just asking if there's some reference which studies operators $A$ with this property: $$\left\langle Ax,x\right\rangle \geq0,$$ for all $x\in H$. And $Ax=0$ whenever ...
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1answer
39 views

Completeness: $\mathcal{l}^2(S)$

Surely, for countable index sets this is just the diagonal trick: $\#S<\infty$ However for arbitrary index sets how do I prove that the limit will actually have only countable non vanishing ...
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25 views

System of equations involving the inner product

I've been reading Ward Cheney's Analysis for Applied Mathematics and he gives the following problem: Indicate how the equation $Ax=b$ can be solved if the operator $A$ is defined by ...
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1answer
15 views

Proof that solution of $\lambda$-affine, linear ODE is entire in $\lambda$

Suppose $F(\lambda)~(\lambda\in\mathbb{C})$ is a linear ordinary differential operator (with, say, domain $D$ dense in some Hilbert space), and is also affine-linear in $\lambda$. Is there a proof ...
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26 views

Formula for trace of particular operators

Let $\mathcal{H}$ be the Hilbert space $L^2(\mathbb{R})$. View the Fourier transform as a unitary operator $\mathcal{F} \in B(\mathcal{H})$. For each function $f \in C_0(\mathbb{R})$, let $T(f) \in ...
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1answer
24 views

Hilbert space (nonseparable): ONB

Every Hilbert space admits an ONB by axiom of choice. For separable Hilbert spaces this can in fact be constructed by Gram-Schmidt. For nonseparable Hilbert spaces there can be no general construction ...
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69 views

Linear and monotone mapping

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuous and monotone, i.e., $$ \left( f(x) - f(y) \right)^\top \left( x-y\right) \geq 0$$ for all $x,y \in \mathbb{R}^n$. Say for which matrices $A ...
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1answer
27 views

help,example about disjoint operators

$T\colon L^2[0,1]→L^2[0,1]$ is given by $$ Tx(t)=∫_0^1 tx(s)\,ds $$ How can we find adjoint operator of $T$ in this space? $\langle Tx,y\rangle= \langle x,T^*y\rangle$ should be okay.But what ...
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18 views

Using derivatives at 0 to define an inner product

Can the following define an inner product on a subspace of the set of functions that are infinitely differentiable on $[-R,R]$. If so, do we get a Hilbert space? $$<f, g> = \sum_{n=0}^\infty ...
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15 views

The relationship between CPTP maps and quadratic forms

Let $H$ be a finite-dimensional Hilbert space (so there is a canonical isomorphism $H\cong H^*$). For a Hilbert space $H$ define $B(H)$ to be the space of linear operators on $H$; we have $B(H)\cong ...
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1answer
18 views

Normal bounded operator

Let $T$ be a bounded normal operator on a Hilbert space. Now I have to show that $T$ is self-adjoint if and only if $\sigma(T) \subset \mathbb{R}$. I already know that for an Abelian unital ...
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1answer
76 views

Dual of $div$ on spaces where the tangential value is fixed

Say $\Omega$ is a domain in $\mathbb R^3$ with a smooth boundary $\Gamma$. Consider the spaces $$ H_{n,0}=\{v\in H^1(\Omega):n\cdot v \bigr |_{\Gamma} = 0\} $$ and $$ H_{t,0}=\{v\in ...
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1answer
70 views

Is there a useful relationship between pointwise and $L^2$ distance?

It would be really convenient to get a bound on the point-wise closeness of functions by knowing their $L^2$ distance. Clearly, if two functions are close in the $L^2$ sense, you cannot get a general ...
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1answer
23 views

Is the image of a closed subspace under a bounded linear operator closed?

This seems obvious, but I can't get the proof straight, and I made up the statement myself, so I'm not sure if it's true in the stated generality. Given a bounded linear operator $T$ in Hilbert space, ...
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24 views

Bounded $L^1$ functions subgroup of $L^2$

I'm currently looking at statistics and the characteristic function. And the claim is that the characteristic function must exist for every probability distribution since every probability ...
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0answers
38 views

Hilbert space without the projection theorem

One succinct statement of the projection theorem in Hilbert space is $A+A^\bot=\scr H$, where $A\in\scr C$, the set of closed subspaces of $\scr H$. (We will also denote the set of all subspaces by ...
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44 views

Prove that if $X$ is a Hilbert space, then $B(X)$ is not a Hilbert space

I`m having a homework question that goes like this: X is a Hilbert space, a complete inner product space, show that B(X) is not a Hilbert space. I`m quite stuck and I would love to understand this ...
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1answer
68 views

Nonseparable $L^2$ space built on a sigma finite measure space

Is it possible to have a nonseparable $L^2$ Hilbert space for which the underlying measure space is sigma finite? I appreciate any example but prefer one built on the Borel sigma algebra of some ...
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1answer
70 views

I dont understand this notation

I`m having a homework question that goes like this: $X$ is a Hilbert space, a complete inner product space, show that $B(X)$ is not a Hilbert space. My only question for now is what does $B(X)$ ...
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21 views

Inner product space or Hilbert space of Quaternionic Functions

In what ways can you define an inner product, $<f,g>$, to create an inner product space or Hilbert space on the set of quaternionic functions $f:\mathbb{H} \rightarrow \mathbb{H}$?
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1answer
36 views

Operator norm of orthogonal projection

I was assigned the following homework problem: "Let $P:\mathcal{H} \to \mathcal{H}$ be bounded and linear. Assume it satisfies $P^2 = P$ and $P^\star = P$. Show $\|P\| \le 1$." This isn't too hard ...
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28 views

The subspace sum of closed subspaces is closed [duplicate]

Given an arbitrary Hilbert space $\scr H$ and closed subspaces $A,B\subseteq\scr H$ with trivial intersection, is it true that $A+B=\{x+y:x\in A,y\in B\}$ is closed? So far, I have the following: Let ...
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1answer
44 views

Strengthened Cauchy-Schwarz inequality

I'm looking for some simple proof of the following consequence of the "strengthened" Cauchy-Schwarz inequality: Let $\mathcal{H}$ be a real Hilbert space such that ...
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12 views

$F_{j_0}=\left\{h:I\to\mathbb{R}/ h(j)=f_j(h(j_0))\ \forall j\not= j_0, h(j_0)\in K\right\}$ compact with supremum norm?

I need very help for the next problem: Let $F=\left\{f_j:\mathbb{R}\to \mathbb{R}/ j\in I, f_j\ continuous,\ and\ equicontinuous\right\}$, I index family. ($f_j$ equicontinuous iff $\forall ...
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2answers
58 views

Find norm of operator

I have a linear functional $$A: L_2[0,2] \to \mathbb R, Ax = \int_0^2(t^2+2)x(t)dt$$ I need to find $C$, trying to measure $C$ and $||Ax||$ to find it, but how can I do it in this problem?
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1answer
33 views

Series convergence in Hilbert space and dual.

I'd like to prove that: $$ \|u_\varepsilon-f\|_*\rightarrow0 \quad\text{in }V^* $$ with $V$ Hilbert and $V^*$ its dual. In particular $u_\varepsilon\in V$. From the precedent points of the proof I ...
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10 views

Showing that $R(T)=R(T^*)$ for a normal operator $T$

For a normal operator $T$ acting on a Hilbert space it is easy to show that the kernel of $T$ coincides with the kernel of the adjoint $T^*$. Thus the norm-closures of the ranges $R(T)$ and $R(T^*)$ ...
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2answers
33 views

Necessity of hypothesis in distance from a set in an inner product space

In Kreyzig's Functional Analysis book, one of theorems in inner product spaces is about the existence and uniqueness of a minimal point from a set. For lack of better means, I have uploaded the ...
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1answer
21 views

Increasing convex-like function in Hilbert space

I am intersted with the differential equation $$x'(t)=f(t,x(t)),\ t\in \mathbb{R}.$$ Can we find an example of a Hilbert space $H$ and a function $f:\mathbb{R}\times H \to H$ which satisfy the two ...
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1answer
35 views

Weak and Norm convergence in Banach Space

I know (and have proven) that in a Hilbert space, $\mathscr{H}$, if a sequence $z_i\overset{w}{\to}z$ and $\|z_i\|\to\|z\|$, then $\|z_i-z\|\to0$. I'm trying to find a counterexample in a Banach ...
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1answer
27 views

How would you define the square of the linear operator

If you define the linear operator norm of $A:X\to Y$ to be $$\|A\|_{op} = \inf\{C>0: \|Ax\|_Y \leq C\|x\|_X \text{ for all } x \in X \}$$ Then how would you define $\|A\|_{op}^2$? My guess is you ...
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2answers
57 views

Want to show that $\{e^{i2\pi nx}:n\in\mathbb Z\}$ form an orthonormal basis for 1-periodic $L^2$ functions.

So here is my problem, I would like to prove that $\{e^{i2\pi nx}:n\in\mathbb Z\}$ form an orthonormal basis for 1- periodic $L^2([0,1])$ functions with respect to, $$\langle ...
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64 views

Proving the completeness of $\mathcal{L}(\mathcal{H})$

Here $\mathcal{L}(\mathcal{H})$ denotes the vector space of all bounded linear operators on a Hilbert space $\mathcal{H}$. We can define a norm on $\mathcal{L}(\mathcal{H})$ by $\|T\| = \inf\{B : ...
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25 views

On a Variational Inequality

Let $H$ be a Hilbert space with real inner product. Consider $f: C \rightarrow C$, where $C \subset H$ is closed and convex. I am not sure about the variational inequality problem: find $x \in H$ ...
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1answer
52 views

Prove vectorspace of bounded functions with supremum-norm is complete and no hilbert space

I have the following: Consider the real vectorspace with bounded functions $$V = \{f:[0,1]\rightarrow\mathbb{R} | \exists C > 0 : f([0,1])\subset[-C,C]\}$$ and the supremum-norm $$||f||_\infty ...
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29 views

give an example of $f \in H^1 (\mathbb R^2)$ [duplicate]

can some help me how to give an example of $f \in H^1 (\mathbb R^2)$ such that $\|f\|_\infty = \infty$. $f\in H^1(\mathbb R^2)$ if $f\in L^2(\mathbb R^2)$ and $f_{x_1},f_{x_2}\in L^2(\mathbb R^2)$ ...
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29 views

Underlying concept to this inner-product “quasi-orthogonal-projection”?

I'm looking at a paper about a finite element method for the LLG-equation. [Bartels/Prohl] In the construction of the scheme they are defining a discrete Laplace operator $\tilde{\Delta}_h$ on the ...
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2answers
49 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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37 views

Bounded Sesquilinear form

Let $X$ and $Y$ be normed spaces. Show that a bounded sesquilinear form $h$ on $X \times Y$ is jointly continuous in both variables.
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27 views

Want to prove an inequality of two norms in a Hilbert space

So here is my problem, Let $D:=[-d,d]\times[-d,d]$ and $C_0^{\infty}$(D) be the set of all smooth functions with compact support in $D$ which are zero on the boundary of $D$. Moreover we have the ...
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1answer
35 views

product of Hilbert spaces

Let $H$ be an infinite dimensional Hilbert space. claim: $H\times H$ with the norm $\|(x,y)\|=\|x\|+\|y\|$ is an Hilbert space. I can't find a counterexample..
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20 views

Easy exercise operators on Hilbert space

Let $(H,\langle\,|\,\rangle)$ be a separable Hilbert space on $\mathbb{C}$. $\rho_{\psi}:=\langle\psi,\,\rangle\psi$, where $\psi\in H$ is such that $\|\psi\|=1$. ...
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2answers
14 views

Orthogonal Projector

Let $(H,\langle\,|\,\rangle)$ be a separable Hilbert space on $\mathbb{C}$. $P_{\psi}:=\langle\psi,\,\rangle\psi$, where $\psi\in H$ is such that $\|\psi\|=1$. I have to prove that $P_{\psi}$ is an ...
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2answers
21 views

Unitary operator on dense set, Unique extension?

given two Hilberspace $H_1$ and $H_2$. Let $V\subset H_1$ and $W\subset H_2$ be dense subspaces. Furthermore let $U: V \rightarrow W$ be an unitary operator. I just want to know whether there is a ...
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18 views

Limit of function of an operator

Let $A_n$ be a sequence of bounded, self-adjoint operators on Hilbert space $\mathcal{H}$. Let us assume that for some vector $\psi\in\mathcal{H}$, $$\lim_{n\rightarrow\infty}A_n\psi = \alpha ...
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1answer
46 views

isometric embedding of l^2

CLAIM: Let $H$ be an infinite dimensional $\mathbb{R}$-Hilbert space. Then the $\ell^2$ sequence space can be embedded in $H$. I think it could be true since every Hilbert space has an orthonormal ...
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21 views

Bounding the distance between $L_\infty$ and $L_2$ for a continuous function

Consider a set of continuous (or even differentiable) functions $f_i(x)$, all defined for $x\in [a,b]$ for $i=1\ldots,N$. Can one define a uniform constant $c$ (which may depend on $f$) such that ...
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1answer
32 views

Is every closed set $K\subseteq \mathbb{C}$ the essential range of a measurable function?

For a complex-valued function $h$ on a measure space $(S,\Sigma, \mu)$, the $\textit{essential range}$ of $h$ is the set of all $\lambda \in \mathbb{C}$ such that for all $\epsilon >0$ the ...
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1answer
41 views

Range and kernel of linear operators

I have a compact linear operator $T$, and I would like to show $$\operatorname{range}(\lambda I-T)=(\ker(\overline{\lambda}I-T^*))^\perp.$$ I have shown the forward inclusion "$\subset$" directly by ...
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2answers
83 views

Compact operator whose range is not closed

I am asked to find a compact operator (on a Hilbert space) whose range is not closed, but I am having trouble coming up with one. My guess is that you need to have some sequence in the range that ...