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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0answers
26 views

$W_0^{1,\:p}(\Lambda)$ is dense in $L^2(\Lambda)$

Let $d\in\mathbb N$ $\lambda$ be the Lebesgue measure on $\mathbb R^d$ $\Lambda\subseteq\mathbb R^d$ be open with $\lambda(\Lambda)<\infty$ $p\ge 2$ $W^1(\Lambda)$ denote the set of weakly ...
-1
votes
0answers
32 views

Possible master thesis [on hold]

I am searching a topic for my master thesis. My interests are especially probability theory (something with brownian motion would be nice) and fourier analysis (also in an abstract Hilbert space ...
9
votes
2answers
657 views

Physical (Quantum Mechanical) Significance of completeness of Hilbert Spaces.

I'm not sure if the question is very 'mathematical',but I'm asking any way. I have a basic knowledge of quantum mechanics and I'm studying Hilbert spaces. I was wondering what is the physical ...
2
votes
2answers
110 views

Proving that a space is complete

There is something that bugs me about the proof I've been shown that $C(\Omega)$ (the space of continuos function on $\Omega$, a compact subset of $\mathbb R^n$) with the $\sup$ norm is complete. ...
0
votes
2answers
37 views

Orthogonal of an Hilbert subspace and density

If $V$ is a subspace of an Hilbert space $H$, I know that the orthogonal of $V$, $V$$^o$, is ($V$closed)$^o$, even if $V$ is not closed. Does this mean that $V$ is always dense in $V$$^o$? Thanks!...
0
votes
1answer
34 views

For two positive operators on Hilbert space is it true that $A \ge B \implies \|A\| \ge \|B\|$?

$H$ is Hilbert space. $A$ and $B$ is positive linear operators from $H$ to $H$ i.e. $\forall x\in H\, (Ax,x),\,(Bx,x)\ge 0$. $A\ge B$ means that $A-B$ is positive. Does that means that $\|A\| \ge \|B\|...
0
votes
2answers
28 views

Question on operator of hilbert space, why $f(x)=\sum_{i}(f|e_i)e_i$?

let $(V,(. |.))$ a Hilbert space. Let $\{e_i\}_{i=1}^\infty $ an orthonormal basis and $f:V\to V$ a linear application. Here are my questions : 1) Why $f(x)=\sum_{i=1}^\infty (f,e_i)e_i$ ? 2) Why ...
3
votes
1answer
67 views

What is the norm of the dual space $H^1(\Omega)'$?

I am working on the Bidomain-Model which, during a time interval [0,T], describes the electrical behaviour of the myocardial muscle considered as $\Omega \subset \mathbb{R}^3$. This model has partial ...
5
votes
1answer
787 views

Showing the basis of a Hilbert Space have the same cardinality

I am trying to show that if we have two orthonormal families $\{a_i\}_{i\in K}$ and $\{b_j\}_{j\in S}$ and these are the basis of some Hilbert Space H, then they have the same cardinality. So If I ...
2
votes
0answers
49 views
+50

Given $Q:ℝ^d→(\text{Hilbert-Schmidt operators }U→ℝ^d)$, find a Hilbert-Schmidt operator $T:U→L^2(ℝ^d,ℝ^d)$ with $Q(x)u=(Tu)(x)$

Let$^1$ $U$ be a separable $\mathbb R$-Hilbert space $d\in\mathbb N$ $\lambda$ denote the Lebesgue measure on $\mathbb R^d$ $\Omega\subseteq\mathbb R^d$ be a bounded domain $H:=L^2(\Omega,\mathbb R^...
1
vote
0answers
60 views
+50

Calculate the trace of $LBB^*$, where $L:H→H$ and $B:=ΦT^{1/2}$ for some $Φ:U_0→H$, an embedding $ι:U_0→V$ and $T:=ιι^*$

Let$^1$ $U$, $V$ and $H$ be $\mathbb R$-Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric $U_0:=Q^{1/2}U$ be equipped with $$\langle u,v\rangle_{U_0}:=\langle Q^{-1/2}u,Q^{-1/2}v\...
3
votes
2answers
59 views

Constructing an $L^2$ space on the unit ring $\mathcal{S^1}$

Revised Question: Starting with $L^2[0,2\pi]$, does the canonical map $$[0,2\pi)\ni\theta\mapsto e^{i\theta}\in\mathcal{S^1}$$(with functions going across in the obvious way) turn $L^2[\mathcal{S^1}]$...
2
votes
1answer
35 views

If $Tv=\mu v$ for some $\mu>0$, then $v\in\ker(T^{1/2})^\perp$

Let $V$ be a separable $\mathbb R$-Hilbert space $T$ be a bounded, linear, nonnegative and symmetric operator on $V$ $(v_n)_{n\in\mathbb N}$ be an orthonormal basis of $V$ with $$Tv_n=\mu_nv_n\;\;\;\...
0
votes
0answers
33 views

If $ι:U→V$ is a Hilbert-Schmidt embedding and $(v_n)_{n∈ℕ}$ is an orthonormal basis of $V$, then $(ιι^*v_n)_{n∈ℕ}$ is an orthonormal basis of $ιU$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle_U)$ and $(V,\langle\;\cdot\;,\;\cdot\;\rangle_V)$ be separable $\mathbb R$-Hilbert spaces $\iota:U\to V$ be a Hilbert-Schmidt embedding $T:=\iota\iota^\ast$ ...
0
votes
1answer
51 views

If $Q$ is an operator on a Hilbert space $U$, $(e_n)$ is an ONB of $U$ consisting of eigenvectors of $Q$, then $(Q^{1/2}e_n)$ is an ONB of $Q^{1/2}U$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ ...
0
votes
0answers
24 views

Extending a unitary operator

Suppose that $\mathcal{H}_1$ and $\mathcal{H}_2$ are two separable Hilbert spaces and that $X\subset \mathcal{H}_1$ is a dense subspace (i.e. $\overline{X}=\mathcal{H}_1$). If $\operatorname{W}:X \to \...
1
vote
2answers
74 views

Quantum Mechanics: position and the separability of Hilbert space?

I would be pleased if someone could point out to me where I go wrong in the following sequence of statements: One model of quantum mechanics identifies states of a particle with normalized vectors (...
0
votes
1answer
40 views

What is the relation between the matrix of a bounded linear operator and that of its adjoint operator?

Let $H_1$ and $H_2$ be finite-dimensional (real or complex) Hilbert spaces, let $T \colon H_1 \to H_2$ be a linear operator, [Then $T$ can be shown to be bounded] and let $T^* \colon H_2 \to H_1$ ...
1
vote
0answers
33 views

How to prove $n(T)=\sup\{|\langle Tx,x \rangle |, \|x\|=1\}$ is a norm on $B(H)$ and $n(T)\lt\|T\|\lt2n(T)$ where $T\in B(H)$? [closed]

Let $H$ be a Hilbert space over $\mathbb C$. If $T\in B(H)$, how to prove that $$n(T)=\sup\{|\langle Tx,x \rangle |, \|x\|=1\}$$ is a norm on $B(H)$ and $$n(T)\lt||T||\lt2n(T)\ \textrm{?}$$ I couldn'...
5
votes
2answers
116 views

Quantum Mechanics state space

In Quantum Mechanics one often deals with wavefunctions of particles. In that case, it is natural to consider as the space of states the space $L^2(\mathbb{R}^3)$. On the other hand, on the book I'm ...
2
votes
1answer
304 views

Sum of closed subspaces of a Hilbert space is closed

Let $M, N ⊂ H$ ($H$ Hilbert), be two closed linear subspaces. Assume that $\langle u, v\rangle = 0$ $∀u ∈ M$, $∀v ∈ N$. Prove that $M + N$ is closed. Take a sequence $(g_n)\in M+N$ such that $g_n\to ...
0
votes
0answers
21 views

weighted shift operator for complex Hilbert space

I am trying to solve that if H is a complex Hilbert space with orthonormal basis $\{e_n\}_{n=1}^{\infty}$ and let $\{a_n\}_{n=1}^{\infty}$ be a sequence with $\lim_{n\rightarrow}a_n = 0$. Define the ...
1
vote
1answer
197 views

the spectral radius of normal operator

Let $H$ be a Hilbert space and $T$ be linear bounded operator in $H$. Prove that if $T$ is normal then the spectral radius of $T$, $$r(T)=\|T\|.$$ Is this TRUE?
0
votes
1answer
45 views

Prove that $\|T\|=\sup_{\|x\|=1}|\langle x,T(x)\rangle|$. [closed]

Let $T$ be a self adjoint bounded linear operator in a Hilbert space $H$. Prove that $$\|T\|=\sup_{\|x\|=1}|\langle x,T(x)\rangle|$$
0
votes
0answers
56 views

A basis for $\text{span}_{L^2(0,1)} \left\{ \eta_\alpha,0<\alpha<1 \right\}$, and how compute coordinates

Let $$\text{span}_{L^2(0,1)} \left\{ \eta_\alpha,0<\alpha<1 \right\},$$ where we take $$\eta_\alpha(t)= \left\{ \frac{\alpha}{t} \right\} -\alpha \left\{ \frac{1}{t} \right\},$$ and $ \left\{ x ...
2
votes
1answer
68 views

Is a Bessel sequence a frame sequence?

$\mathcal H$ being a Hilbert space, $\{g_k\}_{k \in N}$ is a Bessel sequence if there exsits $B >0$ such that $\forall f \in \mathcal H$, $\sum_{k\in N} |\langle f,g_k\rangle|^2 \leq B \| f \|^2$. ...
0
votes
0answers
45 views

$H$ self-adjoint with mass gap, $P \ge 0,\Omega \in D(P)$, $H + \lambda P$ self-adjoint $\implies$ for $\lambda$ small, $H+ \lambda P $ has gap?

Suppose $H$ is a self-adjoint operator on a Hilbert space having a simple isolated least eigenvalue $0$ with gap $1$ ( $H\Omega = 0$, $\Vert \Omega\Vert = 1$ ), $P$ is a non-negative symmetric ...
3
votes
3answers
51 views

Existence of rotations between two points

Let $x,y\in\mathbb R^n$ ($n\in\mathbb N$) be two given points with the same Euclidean norm: $\|x\|=\|y\|$. Does there, in this case, exist an orthogonal matrix $U\in\mathbb R^{n\times n}$ such that $$...
1
vote
1answer
38 views

Generalized polar decomposition

Let $x\in B(H)$. We say $(x,v,y)$ is a polar decomposition for $x$ if, $\bullet$ $y$ is positive. $\bullet$ $v$ is a partial isometry with $x=vy$. $\bullet$ Ker$(x)$=Ker$(y)$=Ker($v$) The polar ...
1
vote
1answer
23 views

Property of inner product on Hilbert space.

Let $H$ be a Hilbert space equipped with inner product $\left< \cdot , \cdot \right>$. Fix $u\in H$ and constant $R_0 > 0$. Define subset $K_u(R_0)$ of $H$ by $$K_u(R_0) =\{w \in H : \left&...
1
vote
1answer
37 views

What does the weak* topology on $\ell_2$ look like?

I am wondering about a way to construct a base or subbase for the weak* topology on $\ell_2$. I am fairly new to topology and functional analysis, so I apologize if the question is not precisely ...
2
votes
0answers
32 views

Sobolev Space with partial inner product

In my work, I encountered the following problem. Consider the set of real-valued functions, which are ``balanced'', that is the set of bounded functions $f(x)$ such that $\lim_{x\rightarrow \pm \...
2
votes
0answers
43 views

Show that usual $L^2$ norm is equivalent to arbitrary norm $||\cdot||$ that satisfies 'convergence condition'

Given $L^2(\mathbb{R})$ consider a norm $||\cdot||$ on $L^2$ such that $(L^2,||\cdot||)$ is a Banach space and every $||\cdot||$-convergent sequence has a subsequence that converges almost everywhere. ...
0
votes
1answer
25 views

Prove or disprove: $\lVert T\rVert=\sup_{\lVert x\rVert=1}|\langle Tx,x\rangle|$, where $H$ is a Hilbert space and $T$ is bdd linear operator.

Edit: To clarify, note that $T:H\to H$. This is a problem on an old preliminary exam in Analysis that I'm working through to prep for my own prelim. My initial thought was to disprove it, but I can'...
0
votes
0answers
55 views

Generalized Absolute Value II

Let $x$ be an operator in $B(H)$. We say a pair $(c,y)$ forms a polar decomposition for $x$ if $y$ is a positive operator, $c$ in $B(H)$ with $x=cy$ such that the restriction of $c$ on $\overline{yH}$...
2
votes
0answers
39 views

Example of Hilbert space non isomorphic to $L2$

I'm looking for an example of a Hilbert space that can't be seen as the countable direct sum of $L^{2}(X,\mu)$ spaces nor subespaces of $L^{2}(X,\mu)$. Some idea to start? Thanks everyone.
7
votes
1answer
2k views

The difference between an isometric operator and a unitary operator on a Hilbert space?

It seems that both isometric and unitary operators on a Hilbert space have the following property: $U^*U = I$ ($U$ is an operator and $I$ is the identity operator) What is the difference between ...
0
votes
1answer
17 views

Sesquilinear forms - How does positiveness imply hermitianity?

In my mathematical methods for physics course notes I find this: A positive sesquilinear form is nondegenerate and Hermitian The first statement is trivial: a ...
2
votes
0answers
31 views

Completeness of 'Hardy Space' $H^2(D)$

Define Hardy Space $H^2(D)$ as a space of holomorphic functions $f$ on unit open disc $D=\{z\in\mathbb{C}:|z|<1\}$ endowed with the norm $$ ||f||^2=\sup_{0<r<1} \int_0^{2\pi} |f(re^{i\...
0
votes
1answer
33 views

Spectrum of two Hilbert spaces

Let $H_1$ and $H_2$ be two Hilbert spaces and $U \in B(H_1,H_2)$ be unitary. Assume that $A\in B(H_2)$ and $B \in B(H_1)$ satisfy $UB = AU$. How can I prove that $sp(A) = sp(B)$ and $sp_p(A) = sp_p(B)?...
0
votes
0answers
13 views

Application of Uniform boundedness theorem: $\langle Tx,y\rangle$ bounded for each $x,y$ then $||T||$ is bounded

For Hilbert Space $X$, if we have a condition on a subset $F\subset B(X)$ ('set of bounded linear operators on $X$') such that $$ \{\langle Tx,y\rangle:T\in F\} $$ is a bounded set for each $x,y\in ...
2
votes
1answer
22 views

Stuck on elementary proof on completeness of $W^{1,2}(\mathbb{R})$ as Hilbert Space

Let $W^{1,2}(\mathbb{R}):=X$ be the space of continuous functions $f$ such that $f\in L^2(\mathbb{R})$ and there exists $f'\in L^2(\mathbb{R})$ such that $$ f(b)-f(a)=\int_a^b f'(t)\,dt $$ for ...
1
vote
1answer
23 views

Characterization of orthogonal projections in terms of operator norms

I want to show the following equivalence: If $X$ is a Hilbert Space and $P\in B(X)$ (i.e. $P$ is bounded and linear) and $P^2=P$, then $$ (\text{im}\,P)^{\perp} =\ker P\iff ||P||\le 1 $$ I know that ...
-2
votes
0answers
30 views

Hilbert Spaces and Hamel Basis

Let $H$ be a Hilbert Space of infinite dimension, $S$ a not finite orthonormal basis and $B$ a Hamel basis to $H$. i) How to show that the cardinality of $B$ is greater than or equal to the ...
2
votes
1answer
49 views

Two orthonormal sets in a Hilbert space. One is complete, the other must be complete.

Given two orthonormal sets $\{e_k\}_{k=1,2\ldots}$, $\{e'_k\}_{k=1,2\ldots}$ in a Hilbert space $H$, which satisfy $$ \sum_{k=1}^\infty \|e_k-e'_k\|^2 < 1. \tag{*} $$ Prove that if $\{e_k\}_{k=1,2\...
1
vote
1answer
37 views

Non-commutaive Gelfand-Naimark theorem and dimension of Hilbert space

It is well known that using non-commutative Gelfand-Naimark theorem for finite dimensional $C^∗$-algebra we can obtain isometric representation on finite dimensional Hilbert space. My question is : ...
0
votes
0answers
20 views

G. Vitali's Result

Let $(x_n)_{n\in\Bbb N}\subseteq\mathcal L_2([a,b])$ be an orthonormal sequence. I want to prove the following: $(x_n)_{n\in\Bbb N}$ is complete $\Leftrightarrow\sum_{n=1}^\infty \big|\int_{[a,t]}...
1
vote
2answers
64 views

Eigenspace and eigenvector inside a Hilbert space

Given $\{v_n\}_{n=1}^\infty$ an orthonormal sequence in a Hilbert space. Let $\{\lambda_n \}_{n=1}^\infty$ a sequence of numbers and $F:H \to H$ defined by $Fx=\sum_{n=1}^\infty \lambda_n \langle x ,...
0
votes
0answers
23 views

Hilbert space mean ergodic theorem application

Let $(u_n)_{n \geq 0}$ be a bounded sequence in a Hilbert space. We define $$ s_h = \limsup \frac 1 N \sum_{n=o}^{N-1} \langle u_{n+h} , u_n \rangle $$ Show that, if $ \lim \frac 1H \sum_{h=o}^{H-1} ...