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

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11 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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35 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
22 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
43 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
63 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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70 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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26 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
58 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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30 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
51 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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30 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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1answer
24 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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18 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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25 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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0answers
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
48 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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2answers
23 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
55 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
95 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 ...
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1answer
52 views

Vanishing of a quadratic form along the orbits of a unitary group

Let $H$ be a (complex) Hilbert space and let $B\colon H\times H\to \mathbb{R}$ be a continuous sesquilinear form (i.e. a continuous function that is linear in one argument and conjugate-linear in the ...
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1answer
28 views

A question about orthogonal projection in Hilbert spaces

Let S be a closed linear subspace of a Hilbert space $H$, and $P_S$ the associated orthogonal projection. I need to verify the following properties. i) $||P_S(x)||\le||x||$ ii)$P_{S^\perp}=I-P_S$, ...
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1answer
25 views

Why is oblique projection not a self adjoint operator?

Why is oblique projection not a self adjoint operator? Here is an explanation of oblique projection.
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1answer
44 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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2answers
69 views

Convergence of sums using Hilbert space techniques [duplicate]

Let $a_n$ be a sequence of real numbers such that $\sum_{n=1}^{\infty} a_nb_n < \infty$ for any sequence $b_n$ satisfying $\sum_{n=1}^{\infty}b_n^2 < \infty$. Prove that ...
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1answer
39 views

Weak convergence of subsequence in Hilbert spaces

Prove that if $x_n$ is a sequence in $H$ (Hilbert space) with $\sup_n||x_n||\le1$, then there is a subsequence $\{x_{n_j}\}$ and an element $x$ of $H$ with $||x||\le 1$ such that $x_{n_j}$ converges ...
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2answers
62 views

Sequence of unit vectors in a Hilbert space

Question: Let $H$ be a Hilbert space and $\{\xi_{i}\}\subset H$ be a sequence of unit vectors. Suppose that $||T_{j}(\xi_{i})-\xi_{i}||\rightarrow0$ as $i\rightarrow\infty$, for $j=1, 2, ...n$ (here ...
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1answer
33 views

Show that a Hilbert space with two inner products has a basis that is orthogonal with respect to both inner products

Let $\mathcal{H}$ be a complex, $n$-dimensional Hilbert space with two inner products $\langle \cdot, \cdot \rangle_1$, $\langle \cdot, \cdot \rangle_2$. Show that there exists a basis $ X = x_1, ...
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51 views

showing uniqueness of a Hahn Banach extension

I am trying to prove the following: If $H$ is a Hilbert space and $G\subseteq H$ is a closed linear subspace, then any bounded linear functional on $G$ has a unique Hahn-Banach extension on $H$. So ...
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1answer
79 views

Proof of an identity in a Hilbert space

Let $x_1,x_2\in H$ be two unit vectors in a Hilbert space $H$ and let $t_1,t_2 : B(H)\rightarrow\mathbb{C}$ be the linear functionals given by $t_j(a) =\langle ax_j,x_j\rangle$. Define ...
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1answer
20 views

Finding the Hilbert Adjoint in this case

If we let $H$ be a Hilbert space with inner product $\langle.,.\rangle$. And we fix $y, z \in H$. Then let $T:H\rightarrow H$ be the bounded linear operator $Tx = \langle x,y\rangle z$. Then what is ...
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1answer
30 views

Compact operator on invariant subspace is compact

Statement: Let $T \in \mathscr{B}(\mathscr{H})$, where $T$ is a compact operator. Let $M$ be a closed invariant subspace of $T$. Show that the restriction of $T$ to $M$ is compact. Attempted Proof: ...
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31 views

Density of a certain subset of $l^2$

let $D$ be the proper set of $l^2$ defined as follows: $$D=\{\{x_k\}| \sum_k |x_k|^2 k < \infty \}$$ How can I prove that this set is dense in $l^2$?
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1answer
21 views

Dimension of intertwining space of unitary representation

I'm currently trying to read through an article by Poguntke, to be found here. The main theorem of the article is the following: Theorem. Let $\pi$ and $\pi'$ be irreducible unitary ...
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29 views

Linear map in Hilbert space.

If you have a linear map $h\mapsto T(h)$ from $H_1$ a real separable space, to Hilbert space $H_2$, it seem that this maps provides an isometry of $H_1$ onto a closed subspace of $H_2$. I try to ...
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1answer
213 views

functions orthogonal to the exponential Bell polynomials

Consider the single variable Bell polynomials $\phi_{n}(x)$ given by: $$\phi_{n}(x)=e^{-x}\sum_{k=0}^{\infty}\frac{k^{n}x^{k}}{k!}$$ I am looking for a set of functions $\tilde{\phi}_{n}(x)$ such ...
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45 views

Is it Hermitian or not?

I have a question, since I'm realy consfused. I am doing quantum information theorem, and there's a theorem that says, that for Hermitian $A\in L(H)$ ($H$ some finite dimensional Hilbertspace) then ...
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1answer
27 views

Let $x\in C^1[-\pi,\pi].$ Prove that $|\int_{-\pi}^{\pi}(x(t).cost-x'(t).sint)dt|^2\le 2\pi\int_{-\pi}^{\pi}(|x(t)|^2+|x'(t)|^2)dt$

Let $x\in C^1[-\pi,\pi].$ Prove that $|\int_{-\pi}^{\pi}(x(t).cost-x'(t).sint)dt|^2\le 2\pi\int_{-\pi}^{\pi}(|x(t)|^2+|x'(t)|^2)dt$ We have no idea to approach the poroblem. Please help??
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20 views

A function $w(t)$ which satisfy $\int dt w(t)F[x](t)=c$ for all x

Consider a differentiable scalar function in two variables $F(x,t)$ for $x\in X$ and $t\in T$, then it can be viewed as an infinite family of scalar functions $\{F[x](t))\}_{x\in X}$. Are there any ...
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1answer
77 views

Two questions on Banach-valued spaces of integrable functions

Let $V$ be a Banach space with dual $V'$ and suppose that $V$ is included into the Hilbert space $H$ so that the inclusion is continuous and dense. Then after identification $H = H'%$ we have $V ...
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1answer
52 views

How to show: $\exists$ a subspace $L$ of a Hilbert space $H$ such that $H\neq L\oplus L^{\perp}$.

How to show: $\exists$ a subspace $L$ of a Hilbert space $H$ such that $H\neq L\oplus L^{\perp}$. Let consider $H=l_2$ where $l_2=\lbrace x=(x_n)^\infty_1: \sum^\infty_1 |x_n|^2<\infty \rbrace $ ...
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21 views

Orthonormal zero Function

I have this exercise Let H be a Hilbert space with orthonormal basis $\{e_n | n\in N\}$ and let $f_n = e_n + e_{n+1}$ If $\langle f,f_n \rangle = 0$ for all $n$ how do I prove that $f=0$ I think i ...
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1answer
33 views

how to show this function doesn't belong to Hilbert space?

I am trying to show $\chi_{B_R(0)}(x) \notin H^1 (\mathbb{R}^n)$ , ∀R>0. since $H^1 (\mathbb{R}^n) := W^{1,2}(\mathbb{R}^n)$ That is, I have to show that $\chi_{B_R(0)} (x) \notin ...
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1answer
41 views

Trace norm of Hermitian matrix

Let $A\in L(H)$ some Hermitian matrix, where $H$ is some finite dimensional Hilbertspace. I want to show $$\left\|A\right\|_{tr} = \max_{U\in U(H)}|\text{tr}(UA)| \ \ \ (*)$$ where U is unitary, and ...
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2answers
59 views

Determining if the span of a set is dense in L^2(0,1)

I am trying to determine whether or not the following statement is true: If $f \in L^2(0,1)$ and $\int_0^1 x^nf(x) = 0$ for all positive integers $n$. Then $f(x) = 0$ I have already verified this ...
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0answers
57 views

Bounded linear functionals on $L^\infty$.

I am looking at a practice final and I am a bit confused by this statement I am trying to prove: "There is a nonzero bounded linear functional on $L^\infty[0,1]$ which vanishes on the subspace ...