Questions tagged [hilbert-spaces]

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

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Schauder basis in $\ell^2(\mathbb{N})$.

I have a question about this problem posted here. Essentially they are trying to prove that the sequence $\{x_n\}_{k = 1}^\infty = \{\alpha e_n + \beta e_{n+1}\}_{n=1}^\infty$ (here $\{e_n\}_{n = 1}^\...
DerpyPenguin's user avatar
1 vote
1 answer
49 views

Differences between real and complex Hilbert spaces

I recently noticed that my go-to reference for stuff related to Hilbert spaces uses the terms "Hilbert space" and "complex Hilbert space" synonymously. Hence the following ...
Filippo's user avatar
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Is the Eigen spectrum of a matrix completely defined by the algebra of its parts?

Consider two vector spaces $\mathbb{C}^n$ and $\mathbb{C}^m$, where $0<n<m<\infty$. Now, I'd like to define matrices $A\in\mathbb{C}^{n\times n}$ and $B\in\mathbb{C}^{m\times m}$ in the ...
Jun_Gitef17's user avatar
5 votes
2 answers
278 views

The projective tensor norm on tensor product of Banach spaces implies the inner product on tensor product of Hilbert spaces?

As presented in the answer of this post, the projective tensor norm on the algebraic tensor product of two Banach spaces $X$ and $Y$ is given by \[ \Vert \omega\Vert_{\pi} = \inf\left\{\sum \lVert x_{...
Keith's user avatar
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Completion of some $C[0,1]$ functions with some inner product is a reproducing kernel Hilbert space

The problem Let $X$ be the space of $C^1[0,1]$ functions with the special property $f(0) = 0$. Consider the inner product defined in the following way: $$\langle f, g \rangle = \int_0^1 f'(x) \...
kodiak's user avatar
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Simple question on linear mappimg $\phi : E \rightarrow F$ between two normed vector space $(E; \| . \|_E)$ and $(F; \| . \|_F)$

Question Let a linear mappimg $\phi : E \rightarrow F$ between two normed vector space $(E; \| . \|_E)$ and $(F; \| . \|_F)$. a) Prove that in $ \mathbb{R}^+ \cup \infty $. We have: $$ \| \phi \| =_{...
OffHakhol's user avatar
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Prove a function from a set of linear operators of a hilbert space to a set of linear operators of a hilbert space is well defined

$\mathscr{L}(\mathcal{H})=$ Set of linear operators from $\mathcal{H}\to \mathcal{H}$. For $T\in \mathscr{L}(\mathcal{H}_A\otimes \mathcal{H}_B)$ specified through $T=\sum\limits_{i,j}\gamma_{i,j}A_i\...
Soham Chatterjee's user avatar
1 vote
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If $(v_n)_n$ is a Hilbert basis for $H$ over $\mathbb{C}$, can you turn $(\mathrm{Re}(v_n), \mathrm{Im}(v_n))_n$ also to a Hilbert basis for $H$?

(Question:) Suppose that $H$ is a Hilbert space over the field $\mathbb{C}$ and that $(v_n)_{n=1}^\infty$ is a Hilbert basis for it, that is a sequence of orthonormal vectors whose span is dense in $H$...
Epsilon Away's user avatar
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if $S \in \mathscr{H}$ and $T \in \mathscr{K}$ then ST is compact. [duplicate]

I understand that $\mathscr{K}(\mathscr{H})$ is a closed subspace of $\mathscr{B}(\mathscr{H})$. Where $\mathscr{K}$ is the compact operators and $\mathscr{H}$ is the bounded operators. I need to ...
Sarah W's user avatar
2 votes
1 answer
56 views

Closest Point With a Property in $L^2([0,1])$

Consider the Hilbert space $L^2([0,1])$ (with Lebesgue measure $dm$). Let $\eta \in L^2([0,1])$ be strictly positive almost everywhere, and essentially bounded, and consider the (nonlinear) functional ...
Joe's user avatar
  • 2,700
1 vote
2 answers
60 views

Norm equality for operators on a Hilbert tensor product

Ok, this could be tough. Suppose you are given bounded linear operators, $\{A_n\}_{n=1}^N\subset\mathcal{B}(\mathcal{H})$ on a Hilbert space $\mathcal{H}$ and $\{B_n\}_{n=1}^N\subset\mathcal{B}(\...
skewfield's user avatar
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Hilbert space valued random variables as a Hilbert space tensor product

Let $\mathcal{H}$ be a Hilbert space and $(\Omega, \mathcal{F}, P)$ a probability space with $L^2$-space $L^2(\Omega)$. I am looking at the space of square integrable $\mathcal{H}$-valed random ...
Snildt's user avatar
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How to show that a space of integrable harmonic functions is a Hilbert space?

From Functional analysis by Peter D.Lax, MR1892228, Zbl 1009.47001, p. 70 If two functions differ by a constant, consider them as equivalent $D$ is a bounded domain in $ \Bbb R^2$. The space $V$ ...
2016's user avatar
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2 votes
1 answer
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Spectral measure and eigenvalues/eigenvectors

Suppose $N$ is a normal operator on some $\mathbb{C}$-Hilbert space $\mathcal{H}$ and let $\mathrm{E}:\mathcal{B}(\sigma(N))\to B(\mathcal{H})$ be the associated spectral measure. Fix $\lambda_0\in\...
Oskar Vavtar's user avatar
1 vote
1 answer
40 views

Filtration of separable Hilbert spaces

Let $\mathcal{H}$ be a separable Hilbert space. Let $(\mathcal{S}_\alpha)_{\alpha\in\Lambda}$ be a decreasing net of closed subspaces of $\mathcal{H}$, i.e. such that for each $\alpha,\beta\in\Lambda$,...
geodude's user avatar
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2 votes
2 answers
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Approximating norm of a Hilbert space point with the norm of a vector

I have the Hilbert space of square integrable functions on $[a, b]$, and what I would like to have is to discretize this space, i.e., find a sequence of finite-dimensional Hilbert spaces $H_K$ of ...
NYG's user avatar
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1 answer
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Solving recurrence relation in Hilbert space.

This is a question that is based off of the book $\textit{An Introduction to Nonharmonic Fourier Series}$ by Young. Assume that $\{e_n\}_{n = 1}^\infty$ is the canonical basis for $\ell^2(\mathbb{N})$....
DerpyPenguin's user avatar
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Show that trace class is a banach space with respect to norm $\|\|_{S_1}$

Show that trace class is a banach space with respect to norm $\| \|_{S_1}$ where the norm is infinite sum of eigenvalues of $(T^*T)^\frac{1}{2}$ and T is a compact operator in a separable hilbert ...
voroshilov's user avatar
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What guarantees that the adjoint of a suitable integral operator, e.g. a Hilbert-Schmidt operator, is again an integral operator with a kernel?

This is likely a silly question, but I was wondering if $T$ is some nice integral transform, e.g. a Hilbert-Schmidt integral operator, with an, say, $L^2(\mathbb{R}^n)$ kernel, what then guarantees ...
Cartesian Bear's user avatar
2 votes
3 answers
107 views

Prove $T$ with bounded basis sum in Hilbert space is compact.

Let $T:\mathcal{H}\rightarrow \mathcal{H}$ be a linear continuous operator between Hilbert spaces and $\{b_i\: |\: i \in I\}$ an orthonormal basis. Prove that if: $$\sum_{i\in I}\lVert T b_i\rVert^2&...
Kadmos's user avatar
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Is the set of polynomials of degree $\le n$ closed in $L^2(a,b)$? [duplicate]

Let $a<b$ be real numbers. I'm asked to prove that for every $f\in L^2(a,b)$ there exists a unique polynomial $p_n$ of degree less than or equal to $n$ such that $$\|f-p\|_2\ge\|f-p_n\|_2,$$ for ...
Little Jonny's user avatar
-1 votes
1 answer
45 views

Why is the spectrum set of the position operator equal to the essential range? [closed]

For $(X,\mu)$ : a $\sigma$-finite measured space, we define $L^2(X)$ : the set of $L^2$-integrable $\mathbb{C}$-valued function on $X$. For $F:X\to \mathbb{C}$ : a measurable function witch is finite $...
neconoco's user avatar
1 vote
1 answer
69 views

A total set of vectors in $\ell^2(\mathbb{N})$

It's been asked to prove that the following set of vectors in $\ell^2(\mathbb{N})$ is total (its span is dense in $\ell^2(\mathbb{N})$): $$ \{c^n := (1,n^{-1},n^{-2}, \ldots) \}_{n=2}^\infty. $$ And ...
Arman Sadeghi's user avatar
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On verifying an inner product on $L^2$; Bridges

In Bridges' Foundations of Real and Abstract Analysis, I'm working an exercise where I believe there's a typo. It's been on my mind for some days now and I'd be grateful for a comment or two. The ...
psie's user avatar
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Showing that $T$ is Hilbert-Schmidt operator

Suppose $H = L_2(B)$ where $B$ is the unit ball in $\mathbb{R}^d$. Let $K(x,y)$ be a measurable function on $B \times B$ that satisfies $|K(x,y)| \leq A|x-y|^{-d+\alpha}$ for some $\alpha > 0$ ...
Grigor Hakobyan's user avatar
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2 answers
61 views

Prove that $ \|x\|_2 = \sup\left \{ x \cdot y \; s.t. \; \|y\|_2 = 1 \right \}$ and that $ x \cdot y = \|x\| $ and $ \|y\|=1$ iff $x = \|x\| y$

In $\mathbb{R}^d$ Question: a) Prove that $ ||x||_2 = Sup\left \{ x \cdot y \; s.t. \; ||y||_2 = 1 \right \}$ b) Prove that $ x \cdot y = ||x|| $ and $ ||y||=1$ iff $x = ||x|| y$ Answer: a) First by ...
OffHakhol's user avatar
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Does $\int_{|z|<\frac{1}{n}} \frac{1}{|z|^{2(d-\alpha)}} \to 0$

I am wondering if $\int_{|z|<\frac{1}{n}} \frac{1}{|z|^{2(d-\alpha)}} \to 0$ as $n \to \infty$ for $z \in \mathbb{R}^d$. It is known that $\int_{\mathbb{R}^d} \frac{1}{|z|^{d-\alpha}}$ is Lebesgue ...
Grigor Hakobyan's user avatar
3 votes
2 answers
94 views

How can you show that the trace class norm $\|A\|_1:=\mathrm{Tr}(|A|)$ satisfies the triangle inequality?

Exact wording of my question is a bit oxymoronic, since a norm by definition is a metric, and thus requires proper context. Let $H$ be a separable Hilbert space over the field $\mathbb{K}$. I am aware ...
Cartesian Bear's user avatar
1 vote
0 answers
78 views

Is $\sup_{x \in S^{n-1}} \vert \langle Ax,x\rangle\vert$ equal to $\sup_{x,y \in S^{n-1}} \langle Ax,y\rangle$ if $A$ is symmetric [duplicate]

Problem: Let $A = (a_{ij})$ be an $n\times n$ matrix. The spectral norm of $A$ is defined as $$\Vert A \Vert = \sup_{x,y \in S^{n-1}} \langle Ax,y\rangle.$$ If $A$ is symmetric, I wonder this norm can ...
Tung Nguyen's user avatar
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Is $GL(\mathcal{H})/\mathbb{C}^* \cong U(\mathcal{H})/U(1)$?

Assume that $GL(\mathcal{H})$ is the group of invertible linear transformations of Hilbert space $\mathcal{H}$ and $U(\mathcal{H})$ is the group of unitary operators on $\mathcal{H}$. We know that $\...
Mahtab's user avatar
  • 437
0 votes
1 answer
30 views

Untiunitary operator on a Hilbert space

A bijective linear (antilinear) operator $A$ on a Hilbert space $\mathcal{H}$ is called unitary (untiunitrary) if $\langle A\psi |A\phi \rangle =\langle \psi |\phi \rangle$ (resp. $\langle A\psi |A\...
Mahtab's user avatar
  • 437
2 votes
1 answer
50 views

Prove that the following function is firmly non expansive

Let $T(x)=\frac{x}{\sqrt{1+x^2}}$ be a function from $R$ to $R$. Prove that the function is firmly non-expansive. Definition: $$||Tx-Ty||^2 + ||(Id-T)x-(Id-T)y||^2 \leq ||x-y||^2$$ If the space is ...
wolf_pack_32's user avatar
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0 answers
86 views

Examples of operator equations on a Hilbert space

Let T be a positive compact operator on a Hilbert space then we know that an operator equation $X^2 = T$ have a solution. Im confused since we have a result which says that a compact operator $T \geq ...
voroshilov's user avatar
1 vote
1 answer
61 views

Limit of sequences in $\ell^2$

Let $\ell^2$ over $\mathbb{C}$ Let $h \in \ell^2$ such that $\forall n \in \mathbb{N}: h_n \neq 0$ where $h_n$ is nth number in $h$ Let $\{v_m\}_{m \in \mathbb{N}}$ be a sequence of points in $\ell^2$ ...
Matey Math's user avatar
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0 votes
0 answers
34 views

weak convergence in an unit ball.

let $x_n \rightarrow x$ weakly and $x_n \in \overline{B(0,1)}$ for all $n$ and $\|x\|_H = 1$ then $\|x_n-x\|_H \rightarrow 0$. We can write $\|x_n -x \|_H^2 = <x,x>+<x_n,x_n> -2Re<x_n,x&...
voroshilov's user avatar
2 votes
0 answers
37 views

Relate a continuous inner product to a discrete one by expanding a kernel around a Dirac delta function?

I am trying to find the discrepancy between an integral and a discrete approximation. The "ideal" inner product between two signals $f(x)$ and $g(x)$ is an integral, $$ \langle f, g \rangle_\...
Alex's user avatar
  • 161
4 votes
1 answer
75 views

If $\{\|T^n\|:n\in\mathbb{Z}\}$ is bounded for an invertible operator $T$ over a Hilbert space, must $T$ be conjugate to an orthogonal operator?

Suppose that $A$ is a invertible matrix with complex entries such that $\{\|A^n\|:n\in\mathbb{Z}\}$ is bounded. We consider the Jordan normal form of $A$. It is well known that \begin{align*} &\...
Jianing Song's user avatar
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1 vote
1 answer
186 views

Show that $\lim_{n \rightarrow \infty}T^nx$ exists for all $x \in H$.

Suppose that $T\geq 0$ is a self-adjoint compact operator with norm $1$, then $\lim_{n \rightarrow \infty}T^nx$ exists for all $x \in H$. Sure with spectral theory for compact operators we obtain the ...
voroshilov's user avatar
0 votes
0 answers
66 views

Kernel mean embeddings in Reproducing Kernel Hilbert Spaces: linear kernel

I'm reading on kernel mean embeddings, and I got stuck a small detail, but I cannot figure it out, so I'm asking it here :) Some context, we are given a Reproducing Kernel Hilbert Space on a compact ...
UpzYaDead's user avatar
0 votes
1 answer
34 views

Proving that $||AB||_{HS}\leq ||B||_H||A||_{HS}$ for Hilbert-Schmidt operator $A$ and a bounded linear operator $B$

The trick I am looking for is likely some well-known trick in functional analysis that I have either not learned or have completely forgotten. I am trying to show that $$||AB||_{HS}\leq ||B||_H ||A||_{...
Cartesian Bear's user avatar
1 vote
1 answer
51 views

Boundedness issue of inverting an infinite Jordan block in $\ell^2(\mathbb Z)$ (shift operator)

I find myself confused about the following thought. Consider the right-shift operator $R$ on $\ell^2(\mathbb Z).$ It has spectrum $\{|z| = 1\}.$ Now pick $\lambda>1.$ Then if $y=(R-\lambda)x,$ we ...
Ma Joad's user avatar
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1 vote
1 answer
34 views

Sum of the spectra of two s-a operators

I am asking something a little more general than my previous question. It seems "trivial" but I cannot find this statement anywhere and I have self-doubts. Let $L_1$ and $L_2$ be two self-...
Gateau au fromage's user avatar
1 vote
0 answers
43 views

Inequality in Hilbert spaces

Prove that $\sqrt{||a+b||^2+||c+d||^2} \leq \sqrt{||a||^2+||b||^2} + \sqrt{||c||^2+||d||^2}$, when $a,b,c,d$ are vectors in a Hilbert space $H$. My attempt: $$ ||a+b||^2=||a||^2+||b||^2+(||a+b||^2-||...
wolf_pack_32's user avatar
0 votes
0 answers
34 views

On the projective Hilbert space

Let $\mathscr{H}$ denote a Hilbert space. The projective Hilbert space is defined as $\mathbb{P}\mathscr{H}:=(\mathscr{H}\setminus \{ \mathbf{0}\})/\mathbb{C}$. It is the set of one-dimensional ...
Mahtab's user avatar
  • 437
1 vote
1 answer
52 views

help showing a property for a weak operator closed $^*$-subalgebra of the bounded operators of a Hilbert space.

Let $A$ be a weak operator closed $^*$-subalgebra of the bounded operators of a Hilbert space. If $T \in A$, then I am trying to show that $P_{(\ker T)^\perp} \in A$(projection for $(\ker T)^\perp$. ...
The Unique Operator's user avatar
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0 answers
20 views

Diagonalizing Block Matrices of Hilbert Space Operators

I’m running into difficulty computing operator exponentials for “block matrices” of Hilbert space operators. It would be extremely useful to be able to diagonalize these block matrices. While the ...
Joe's user avatar
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0 votes
1 answer
44 views

Is self-adjoint the strong limit a sequence of seft-adjoint operators on dense set?

Consider $A_n$ to be a sequence of self-adjoint operators on a separable Hilbert space $H$. Moreover, assume that $A_n$ is a bounded operator for any $n$, and that $A_nx$ converges strongly to $Ax$ ...
NessunDorma's user avatar
3 votes
1 answer
116 views

Hilbert space under a mean value inner product

I am looking for a (Hilbert) space of (real-valued) functions on $\mathbb{R}^n$ where the following map defines an inner product: $$ (f,g) \mapsto \lim_{T\to\infty} \frac{1}{\mathrm{Vol}[-T,T]^n}\int_{...
Bram's user avatar
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0 answers
29 views

Continuous and compact embedding between seperable Hilbert spaces is always dense?

What could be an example of a continuous, compact embedding betweend two seperable Hilbert spaces $H$ and $U$, which is not dense? i.e. is there a map $i: U\rightarrow H, x\mapsto x$ which is not ...
Furkan's user avatar
  • 69
1 vote
1 answer
29 views

Can I perform finite-dimensional inner products with functions as vector components?

I'm an engineer. I'm struggling with infinite-dimensional linear spaces and would appreciate some help. I will describe my question as an example because I'm unfamiliar with the subject. Let's say $u(...
Marcelo's user avatar
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