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

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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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1answer
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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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
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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2answers
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
42 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
84 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
50 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
26 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
41 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
66 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
35 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
32 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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2answers
46 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
77 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
29 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
20 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
39 views

Sufficient condition for compactness of an operator [closed]

Let $A$ be a bounded linear operator from a Hilbert space $H$ to itself such that for each orthonormal basis $\{e_n\}$ of $H$ we have $\langle Ae_n , e_n\rangle \rightarrow 0$. Then show that $A$ is ...
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1answer
199 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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2answers
44 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
51 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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2answers
20 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
39 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
49 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
53 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 ...
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1answer
43 views

Applying perp twice in a hilbert space

Let $H$ be a hilbert space and let $K \subset H$ be a subspace. Then $\overline{K} \subset K^{\perp\perp}$, but why does the reverse inclusion hold?
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1answer
35 views

Show compactness of an operator with Arzelà–Ascoli

We have $K\colon L^{2}(a,b) \rightarrow L^{2}(a,b)$ such that $ Kf(t)=\sum_{j=1}^{n}\phi_{j}(t) \int_{a}^{b} \psi_{j}(S) f(s)ds$ where $\phi_{j} ,\psi_{j} \in L^{2}(a,b)$. We want to show that K is ...
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1answer
23 views

Nonexpansive Affine Operators in Hilbert spaces

Let $H$ be a Hilbert space with real inner product $\langle \cdot, \cdot \rangle : H \times H \rightarrow \mathbb{R}$. Let $T$ be an affine operator. Show that $T$ is nonexpansive, i.e., $\left\| ...
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1answer
79 views

Invertible iff Bounded below and dense range

Statement: Given a Hilbert space $\mathscr{H}$ and $\mathscr{K}$ and a bounded operator $A \in \mathscr{B}(\mathscr{H}, \mathscr{K})$. Show that $A$ is invertible if and only if $A$ is bounded below ...
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0answers
26 views

Convex Feasibility problem on Hilbert space

Let $H$ be a Hilbert space with real inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$. Consider a closed convex set $X \subset H$ and let $P_X: H \rightarrow X$ be the ...
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127 views

Projection and Pseudocontraction on Hilbert space

Let $H$ be a real Hilbert space with inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$, and induced norm $\left\| \cdot \right\|: H \rightarrow \mathbb{R}_{\geq 0}$. Let ...
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1answer
48 views

Hilbert space inequality $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$

In prelim prep I came across 'given $\epsilon$ there exists $C_{\epsilon}$ such that $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$. It is asserted without proof, so I've tried ...
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26 views

Does Hilbert space with countable dimensions exist? [duplicate]

If there is a Hilbert space with infinite dimensions, can it have countably infinite dimensions? And does Banach space with countable dimensions exist?
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24 views

Limit of exp of self-adjoint operator

Let $A$ be self-adjoint (possibly unbounded) operator on Hilbert space $\mathcal{H}$. Under what conditions $w-\lim_{t\rightarrow\infty} e^{i A t}=P_0$, where $w-\lim$ - the limit in weak operator ...
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21 views

Hilbert-Schmidt theorem

In the Hilbert-Schmidt theorem what it means : $A e_n=\lambda_n e_n$ ? Thank you .
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1answer
39 views

Need help proving the equivalence of two norms !

Hey I could use alot of help with this problem please! Let (X, <-,->) be a Hilbert space over R. Then, let A: X -> X be a linear operator. Suppose that A is symettric and positive definite. Let ...