Tagged Questions

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

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2
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1answer
35 views

Hilbert Spaces - an application of the minimax principle.

Let $A$ be a compact, self-adjoint operator, $A \geq 0$. We need to prove that for any orthonormal system $\{e_i\}_1^{\infty}$ and for any $N$, $$\sum_1^N \langle Ae_i,e_i \rangle \leq \sum_1^N ...
5
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0answers
38 views

Two Hilbert spaces $V \subset H$, a basis for both spaces?

Let $V \subset H$ be a pair of Hilbert spaces (with different inner products). The embedding is continuous and dense, and both spaces are separable. Is it always the case that one can I find a ...
1
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1answer
31 views

Hilbert Spaces; eigenvalues of $PBP$ vs. $B$ for $B$ compact selfadjoint and $P$ orthoprojection.

An exercise I have come upon while studying Hilbert Spaces: Let $A$ be a compact operator, and $P \in L(H)$ be an orthoprojection. Prove that $$\lambda_n (PA^*AP) \leq \lambda_n (A^*A)$$ (Where ...
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3answers
21 views

Describing a Subset of a Hilbert Space $H$

Let $H$ be a Hilbert space. How can we describe the set $\{ x \in H \mid \|x-y\| = a \|x-z\| \},$ where $y, z \in H$ are fixed and $a > 0$? Geometrically how does it look like?
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2answers
55 views

Spectral Measures: Support vs. Concentration

The support of a Borel spectral measure is defined by: $$\lambda\in\mathrm{supp}E:\iff E(U)>0\quad\lambda\in U\in\mathcal{T}$$ (See the german wikipedia article: Spektralmaß) Now, consider a Borel ...
0
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2answers
42 views

Spectral Measures: Property

Given a Hilbert space $\mathcal{H}$ and spectral a measure $E:\Sigma(\Omega)\to\mathcal{B}(\mathcal{H})$. Can you give me a hint for: $$E(A)E(B)=E(A\cap B)$$ So far for disjoints I checked: ...
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1answer
64 views

Spectral Measures: Integration of Product

Given a Hilbert space $\mathcal{H}$ and spectral a measure $E:\Sigma(\Omega)\to\mathcal{B}(\mathcal{H})$. Define the integral of simple functions by: $$\int_\Omega ...
2
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1answer
51 views

Why are “not bounded” operators not everywhere defined?

Let $X, Y$ be Banach spaces, $\mathcal{D}(T)$ a subspace of $X$, and $T\colon X\to Y$ a linear map. Such a $T$ is commonly called an unbounded linear operator, where unbounded just means that the ...
1
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1answer
37 views

Estimating the modulus of continuity of translation in $L^2$ by a Sobolev norm of the function

For any $s\in \mathbb{R}$ define the Hilbert space $H^s(\mathbb{T})$ by means of norm $$\|f\|^2_{H^s}=|\widehat{f}(0)|^2+\sum_{n\in\mathbb{Z}}|n|^{2s}|\widehat{f}(n)|^2.$$ Show that for any $0\leq ...
2
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2answers
44 views

left regular representation of SU(2)

in Sepanski's book Compact Lie groups, he describes the representation theory of SU(2) as being isomorphic to $\mathbb{N}$ (SU(2) acts irreducibly on the (n+1)-dimensional space of homogeneous ...
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1answer
13 views

Integration over subsets of the complex plane.

Original Problem: Let $\Omega\subset \mathbb{C}$ be an open set and let $f:\Omega\to\mathbb{C}$ be holomorphic such that $f\in L^{2}(\Omega)$. Show that if $B(z,r)$, the ball of radius $r$ ...
0
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1answer
27 views

Existence of minimum norm solution to linear equation $Tx =y$

Let $T: X \to Y$ be a bounded linear map between Hilbert spaces $(X, \langle \cdot , \cdot \rangle_X)$ and $(Y, \langle \cdot , \cdot \rangle_Y)$ (the Hilbert spaces may be complex or just real ...
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2answers
68 views

Exercise 23 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”

Consider exercise 23 from chapter 4 ("Hilbert Spaces: An Introduction") of [1] (p. 198). Any help will be much appreciated. Thank you in advance. Suppose $\{T_k\}$ is a collection of bounded ...
1
vote
1answer
21 views

Extending mappings on simple tensors

Consider the following situation: Let $H, K$ be Hilbert spaces and let $\Phi$ be some mapping defined on simple tensors in $H\otimes K$ taking values in $B(H\otimes K)$ with the property that each ...
0
votes
1answer
45 views

Inner Product in Hilbert Space

Let $H$ be a Hilbert space and $\phi_{1}, \dots, \phi_{n} \in H$ are linearly independent vectors. How can we construct the inner product on $H$ such that $\phi_{1}, \dots, \phi_{n}$ become orthogonal ...
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0answers
17 views

Is a complex function really just an infinite dimensional matrix?

I have recently sort of come to the understanding that integrating two functions multiplied together is a sort of infinite dimensional dot product, and I only know this from taking an undergraduate ...
2
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1answer
63 views

Exercise 31 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”

The following problem shows that $\{e^{inx}\}_{n \in \mathbb{Z}}$ is an orthonormal basis of $L^2([-\pi, \pi])$. It is taken from [1] (exercise 31 of chapter 4: "Hilbert Spaces: An Introduction", pp. ...
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1answer
34 views

Are there any interesting Hilbert spaces that do not present as function spaces?

I was pondering this question in class earlier: All separable, infinite dimensional Hilbert spaces are isometrically isomorphic. Thus, in particular, any such space is isometrically isomorphic to ...
2
votes
2answers
91 views

Exercise 34 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”

Consider exercise 34 from chapter 4 ("Hilbert Spaces: An Introduction") of [1] (p. 201): Let K be a Hilbert-Schmidt kernel which is real and symmetric. Then, as we saw, the operator $T$ whose ...
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0answers
39 views

Problem 8 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein and Shakarchi's Real Analysis

The following is problem 8 from chapter 4 ("Hilbert Spaces: An Introduction") of Stein and Shakarchi's Real Analysis. Suppose $\{t_k\}$ is a collection of bounded operators on a Hilbert space $H$. ...
0
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0answers
24 views

Relation between two spectra

This seems like an easy enough computation but I'm stuck! Let $X \in B(H)$ for a Hilbert space $H$ such that $X^{2}=0$, but $X\neq 0$. With respect to the decomposition $H=\text{ker}X \oplus ...
2
votes
1answer
46 views

“Almost” Hilbert spaces

This question is a bit (very?) vague. Is there some notion of how "close" a Banach space is to being a Hilbert space? What I have in mind is something like a real or complex valued function on ...
1
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1answer
45 views

How do you prove a hilbert transform?

I am stuck with this question below, I need help;
2
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0answers
18 views

Show that the set $\{e^{\pm i (n-1/4)t}: n=\pm 1,\pm2,\pm3,\ldots\}$ is not a basis for $L^2[\pi,\pi]$

Show that the set $\{e^{\pm i (n-1/4)t}: n=\pm 1,\pm2,\pm3,\ldots\}$ is not a basis for $L^2[\pi,\pi]$. (HINT: The series $$\sum_n c_n e^{i\lambda_n t}$$ with $\lambda_n=n-1/4$, diverges in ...
2
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1answer
31 views

Inner product in Besicovitch space

Besicovitch space is a space constructed in the following way: We take the closure (with respect to the uniform convergence topology) of a linear span: ...
3
votes
1answer
27 views

Operator norm and Hilbert Schmidt norm

I'm looking for a proof of \begin{equation} ||T||\leq ||T||_{HS}, \end{equation} for which it is sufficient to show \begin{equation} ||Tx|| \leq ||x|| \cdot ||T||_{HS} \forall x\in H, x\not=0 ...
2
votes
1answer
32 views

Prove that the space P is a Hilbert Space.

Prove that the space P of all entire functions of the form $$f(z)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\varphi(t) e^{-izt} dt,$$ is a Hilbert Space, where $\varphi\in L^2[-\pi,\pi]$. The inner product of ...
2
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0answers
42 views

Distance to a closed subspace of a Hilbert space in terms of inner product with the unit normal

Let $M$ be a closed subspace of a Hilbert space $H$, and suppose $x_0\in H$ Show that: $$\min(\|m-x_0\|, m\in M)=\max(|\langle x_0,n\rangle|, n\in M^\perp ,\|n\|=1)$$ I know that $|\langle ...
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0answers
17 views

Does a set of 'm' linearly independent continuous functions constitute a Hilbert Space

If I have a Sobolev space $\mathcal{H}^m[a,b]$ of functions $f : [a,b]\rightarrow\mathbb{R}$ where for all $f \in\mathcal{H}^m[a,b]$, $f$ and all derivatives up to order $m-1$ are absolutely ...
1
vote
1answer
42 views

Show an operator is compact if $\sum \|Te_n\| < \infty$

Let $H$ be a separable Hilbert space, define a bounded linear operator $T:H \rightarrow H$, show it is compact if $\sum \|Te_n\|_H < \infty$. My attempt: We show that $T(B)$ is totally ...
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1answer
40 views

Non-separable Hilbert spaces in duals

A topological space $X$ satisfies the countable chain condition if every family of pairwise disjoint open sets in $X$ is countable. I am looking for a reference to the following fact: Suppose that ...
1
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1answer
32 views

Exercise about an operator (adjoint and spectrum)

Let $y\in c_0$ and define the operator from $l^2 \rightarrow l^2$ as the following $$T\bigg(\sum x_n e_n\bigg) \mapsto \sum y_n x_n e_n.$$ I have shown that the operator is continuous, compact and ...
0
votes
2answers
49 views

Spectrum of self-adjoint operator on Hilbert space real

My book says that a self-adjoint bounded linear operator $A:H\to H$ on a complex Hilbert (not sure if separability is needed) space has a real spectrum. I guess that the key is in the fact that any ...
3
votes
2answers
51 views

How to show that $\|a+b+c\|^2\leq 3\|a\|^2+3\|b\|^2+3\|c\|^2$

Show that $$\|a+b+c\|^2\leq 3\|a\|^2+3\|b\|^2+3\|c\|^2$$ where $a,b,c$ are in some Hilbert space $(H,\langle\cdot,\cdot \rangle)$? I see that we have $$\|a+b\|^2\leq2\|a\|^2 +2 \|b\|^2$$ due to the ...
0
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0answers
42 views

Invertibility of an operator involving inner product

Let $H$ be a Hilbert space with basis $b_i$. For all $t$, let $f(t;\cdot,\cdot)$ be an inner product on $H$. For each $j$, is $$\int_0^T \sum_{i=1}^\infty f(t,b_i,b_j)x_j(t)=0$$ uniquely solvable for ...
0
votes
1answer
22 views

Continuous Quadratic Form $\implies$ Continuous Sesquilinear Form

Given a Hilbert space $\mathcal{H}$. Consider a quadratic form $q:\mathcal{H}\to\mathbb{C}$. Define its inducing sesquilinear form: $$s:\mathcal{H}\times\mathcal{H}\to\mathbb{C}: ...
0
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1answer
22 views

Infinite-dimensional version of Gram matrix is invertible

We all know that a Gram matrix (a matrix with entries that are inner products of basis functions) is a invertible. Suppose I have $a_{ij} = (h_i, h_j)_H$ where the $h_j$ are basis functions of a ...
0
votes
1answer
40 views

Isomorphism between Euclidean space and its conjugate

I know that, if $H$ is a Hilbert space, for any continuous linear functional $f\in H^{\ast}$ there is a unique element $x_0\in H$ such that $\forall x\in H\quad f(x)=\langle x,x_0\rangle$. Moreover, ...
1
vote
3answers
43 views

Inner product space over generalized number systems

Apologies for the lengthy setup, but I want to make sure I am clear on how I am using the notation, and what I mean by the phrase "generalized number system". Define a generalized number system $G$ ...
2
votes
1answer
41 views

Eigenfunctions and spectrum of $T:H \to H^*$ where $H$ is a Hilbert space

Let $H$ be a Hilbert space with dual $H^*$. Suppose $T:H \to H^*$ is a linear bounded symmetric operator. (We probably don't want to identify $H$ with $H^*$). Can we talk about the ...
3
votes
1answer
42 views

The strong operator limit of a sequence of unitary operators

If $\mathcal H$ is a Hilbert space and $U_n \in B(\mathcal H)$ is a strong-operator convergent sequence of unitary operators, say $U_n\rightarrow U$, is it true that $U$ is unitary? More explicitly, ...
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0answers
16 views

What is the closest self-adjoint (positive) operator to a given operator?

Given an operator $\rho$ on a Hilbert space $H$, is there a notion of nearest self-adjoint (positive) approximation of $\rho$ for a suitable norm? More specifically, does there exist an algebraic ...
0
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3answers
61 views

Selfadjoint Operator: Empty Spectrum

Can a selfadjoint operator have empty spectrum? (As far as I remember, yes; but just to be sure.) The point is that if so then the closure of its spectrum cannot equal the convex hull of its ...
1
vote
2answers
31 views

Can we show that $E\|X-Y\|^2 \leq E\|X-Z\|^2 + E\|Z-Y\|^2$

Let $X,Y,Z$ be some random elements on some Hilbert space $(H,\langle\cdot,\cdot\rangle)$. Can we show that $$E\|X-Y\|^2 \leq E\|X-Z\|^2 + E\|Z-Y\|^2$$ I can clearly see that $$E\|X-Y\|^2 \leq ...
5
votes
1answer
95 views

Continuous linear image of closed, bounded, and convex set of a Hilbert Space is compact

Is my proof of this proposition correct ? And is this proposition well known? Proposition: Let $C$ be a closed, bounded, and convex set in a separable Hilbert space $H$. Let $L : H \to \mathbb{R}^n$ ...
2
votes
2answers
43 views

Is the distance between disjoint closed convex subsets of a Hilbert space positive? Is it attained?

Let $H$ be an infinite dimensional and separable Hilbert space. Let $A,B$ be infinite, closed and convex subsets of $H$. If $A$ and $B$ are disjoint and if at least one of them is bounded, is the ...
1
vote
1answer
18 views

Commutant of a set of operators and norm topology.

In the references I have it's remarked that the commutant $S'$ of a set $S$ in $B(H)$, where $H$ is a Hilbert space, is closed in the weak operator topology. And this is true because if ...
0
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0answers
34 views

Hilbert subspaces of $B(\mathbb{R}^n)$

Apart from the one-dimensional subspaces, what are the Hilbert subspaces of $B(\mathbb{R}^n)$? I'm not even sure if such subspaces exist.
2
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0answers
44 views

Category of Hilbert Spaces

Is it possible to triangulate the category of Hilbert spaces and bounded linear operators? I assume that one candidate for triangulation is the double dual space. What is a fact is that this ...
3
votes
1answer
49 views

Inequalities with $\|x-y\|$, $|\langle x,y\rangle|$, and $\sqrt{\|x\|^{2}+\|y\|^{2}}$ in a Hilbert space

Let $H$ be a Hilbert space, and let $\|x\|$ denote the norm of $x\in H$, and $\langle x,y\rangle$ denote the inner product of $x,y\in H$. For $x,y\in H$ let us denote $\alpha(x,y)=\|x-y\|$, ...