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

learn more… | top users | synonyms

3
votes
0answers
69 views

Spectral Measures: Helffer-Sjöstrand

Problem Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard almost analytic extensions: $$f_E\in\mathcal{C}^\infty_0(\mathbb{C}):\quad ...
3
votes
0answers
51 views

Weak vs strong convergence for unitary operators

Suppose $H$ is a separable complex Hilbert space with inner product $(\cdot,\cdot)$ and norm $\|\cdot\|$, where $\|u\|^2 = (u,u)$. Suppose $u, u_1, u_2, \dots \in H$. Then $\lim_{n \to \infty} u_n = ...
3
votes
0answers
41 views

Spectral definition of (fractional) Laplacian, need help understanding text

Let $\varphi_k$ and $\lambda_k$ be the eigenfunctions and eigenvalues of the Dirichlet Laplacian $-\Delta$ on some bounded domain $\Omega$. We know $\varphi_k$ are smooth and form an orthogonal basis ...
3
votes
0answers
67 views

An inequality for positive operators

Let $S$ and $T$ be positive operators on a Hilbert space $\mathcal{H}$. Suppose that $S \le T$. Since the square root function is operator monotone, it follows that $S^{1/2} \le T^{1/2}$. Does the ...
3
votes
0answers
105 views

Proof of the Riesz-Schauder Theorem (for compact operators) using the Analytical Fredholm Theorem

First of all sorry for my bad English, I'm an Italian student, hope to let you understand! I'm having a little troubles with the proof of the Riesz-Schauder theorem for Compact Operators. Some infos ...
3
votes
0answers
82 views

Approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets?

Are there some common ways to approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets? (note that the unit ball isn't compact.) My goal is to prove a statement which holds ...
3
votes
0answers
32 views

Limits of trajectory of gradient flow in Hilbert space

I have been studying about gradient flow in Hilbert space of a Morse function $f$. Specifically, let $X$ be a Hilbert space and $f : X\to \mathbb R$ be $C^3$ function. The gradient flow here is ...
3
votes
1answer
117 views

Inequivalent norms (given by different inner products) on infinite dimensional Hilbert space.

I have this question in reviewing for my exam. Let $H$ be an infinite dimensional Hilbert space. Write down an inner product on $H$ that gives a norm inequivalent with the original norm. Is $H$ ...
3
votes
1answer
47 views

Different norm on $\ell_p$-space and Hilbert space

We define $\ell_p=\{(x_n)_{n\in{\mathbb{N}}}\in\mathbb{C}^\infty:\sum_n{|x_n|^p}<\infty\}$. With the usual usual norm $||.||_p$ this becomes a Bancach space. Also we have the usual inner product : ...
3
votes
0answers
56 views

Sum of closed linear subspaces necessarily closed?

Let $H$ be an infinite-dimensional Hilbert space. Let $L_1,L_2 \subset H$ be two closed linear subspaces. If it is also known that $L_1 \perp L_2$ then it is not hard to show that $L_1 + L_2 = \{x_1 ...
3
votes
1answer
56 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 $\alpha(x,y)=\|x-y\|$, ...
3
votes
0answers
32 views

The relationship between CPTP maps and quadratic forms

Let $H$ be a finite-dimensional Hilbert space (so there is a canonical isomorphism $H\cong H^*$). For a Hilbert space $H$ define $B(H)$ to be the space of linear operators on $H$; we have $B(H)\cong ...
3
votes
1answer
169 views

Showing that the space of Hilbert-Schmidt operators form a Banach space.

How do i show that the set of Hilbert-Schmidt operators $HS(H) = \{T \in B(H) \; : \; \sum^{\infty}_{n=1}\|Te_n\|^2 < \infty \}$ for some countable ONB $\{e_n\}$, on a separable Hilbert Space ...
3
votes
0answers
82 views

Does a “typical” reproducing kernel on a manifold generate an infinite-dimensional RKHS?

Let $M$ be a finite dimensional manifold and $k:M\times M\rightarrow\mathbb{R}$ a smooth reproducing kernel. Is the set of all such $k$ with the property that the reproducing kernel Hilbert space ...
3
votes
0answers
118 views

Proving that a certain differential operator is self-adjoint

Consider the differential operator $T:u\mapsto -iu'$ for any $u\in D(T):=\{f\in AC[-\pi,\pi]~|~f'\in L^2(-\pi,\pi),f(-\pi)=f(\pi)\}$; we consider $T$ as a densely-defined operator on $L^2(-\pi,\pi)$. ...
3
votes
0answers
94 views

An orthonormal basis for a Hilbert space

Can anyone give me some hint on the following problem without using any knowledge about complex analysis or Fourier analysis? Thanks a lot! Consider the Hilbert space $$\mathscr{H}:=\bigg\{f\text{ ...
3
votes
0answers
72 views

What is $H^1([0,1]) \otimes H^1([0,1])$?

Let $H^1([0,1])$ denote the Sobolev space $H^1$ on the interval $[0,1]$. What is $H^1([0,1]) \otimes H^1([0,1])$? Here, $\otimes$ the tensor product of Hilbert spaces. In particular, how is that ...
3
votes
0answers
77 views

Is this operator bounded?

Let $w_j$ be a basis ( not orthogonal) of the Hilbert space $H$. For $h = \sum^\infty a_iw_i$ define $P_n(h) = \sum_{i=1}^n a_iw_i$. Is this operator bounded in $H$ I don't think it is but I feel ...
3
votes
2answers
220 views

Proof of the Riesz Representation Theorem

Theorem: Let $F$ be a continuous linear functional on the Hilbert space $H$, then $\exists !$ (exists one and only one) $y \in H$ such that $F(x) = (x,y)$ for $x\in H$. Proof: Uniqueness: ...
3
votes
0answers
200 views

Prove this equality in a Hilbert Space $H$ with Riesz's Representation Theorem.

For $x \in H$, prove that $\sup_{\|z\| = 1} \langle x , z \rangle= \| x \|$ Here is a quick one. If someone could improve this it would be great Proof By Cauchy Schwarz, $\langle x,z \rangle ...
3
votes
0answers
233 views

Prove Heisenberg uncertainty principle (measure and integration theory)

Here is a question in measure and integration theory, Let $f$ be a continuously differentiable complex function on $\mathbf{R}$ s.t. the functions $x \mapsto xf(x)$ and $f'$ are in ...
3
votes
0answers
79 views

Estimate finite-rank operator

I have the the following problem. Let $H$ be a Hilbert space with orthonormal basis $(e_{j})_{j\in \mathbb{N}}$. Let $x\in [a,b]$, for all $h\in H$ $$ (Bh)(x) = \langle h,k_{x} \rangle,$$ with ...
3
votes
1answer
30 views

Confusion related to reproducing kernels

I was reading this paper and I came across Reproducing Kernel Hilbert Space. I tried to read some references related to it. However, I couldn't understand much. I didn't get why they are called ...
3
votes
0answers
63 views

A set of trajectories as a linear subspace of Hilbert space

Let $\left\{S(t)\right\}_{0 \leqslant t \leqslant \theta}$ be a strongly continuous semigroup of linear continuous operators in Hilbert space $H$, $S(0) = I$. Let $x$ be some element of $H$. Then its ...
3
votes
0answers
171 views

Must-read papers in Operator Theory

I have basically finished my grad school applications and have some time at hand. I want to start reading some classic papers in Operator Theory so as to breathe more culture here. I have read some ...
3
votes
0answers
53 views

Existence of an ergodic-looking limit in a Hilbert space

This is part of a problem from Reed & Simon's Functional Analysis -- I'll write the problem first. Let $V$ be a linear transform on the Hilbert space $H$, such that its powers are uniformly ...
3
votes
1answer
113 views

What is $\mathcal{C}(S^{1})$? (Where $S^1$ denotes unit circle)

What is $\mathcal{C}(S^{1})$ (Continuous function on a unit circle)? (Where $S^1$ denotes unit circle) I saw this in a proof of showing Fourier Basis $S:=\{1,\sqrt{2}\cos{nx},\sqrt{2}\sin{nx}\}$ is ...
3
votes
0answers
169 views

Is there a deeper connection between the two Riesz's Representation Theorems?

I have been reading Kreyszig's Functional Analysis when I encountered two versions of Riesz's Representation Theorems: (1) Every bounded linear functional $f$ on a Hilbert space $H$ can be ...
3
votes
0answers
279 views

How does the parallelogram law imply the existence of an inner product for a given norm? [duplicate]

Possible Duplicate: Norms Induced by Inner Products I am trying to prove to that if a norm of a vector space satisfies the parallelogram law ($\| \vec x + \vec y \|^2 + \| \vec x - \vec ...
3
votes
0answers
90 views

Counting balls in Hilbert spaces

Let $W$ be a real Hilbert space of dimension $n$ and $V$ a Hilbert subspace of dimension $m$. Assume that $f_1,\cdots,f_k$ are points in $W$ such that the following holds: there exists $\sigma>0$ ...
2
votes
3answers
2k views

Canonical examples of inner product spaces that are not Hilbert spaces?

That is, what are some good examples of vector spaces which are inner product spaces but in which not every Cauchy sequence converges?
2
votes
2answers
2k views

A proof of the Riesz representation theorem

I'm having trouble filling the steps in this guided proof of Riesz's representation theorem. (I already have a proof I can understand, but I'd like to understand this one too.) Let $H$ be a Hilbert ...
2
votes
3answers
865 views

Orthonormal basis in $L^2(\Omega)$

In the one dimension case, where $\Omega\subseteq{\bf R}$ is a bounded domain, for example $\Omega=[0,2\pi]$, one can find the orthonormal basis for $L^2(\Omega)$: $$\{e_n\}_{n\in {\bf Z}}$$ where ...
2
votes
1answer
131 views

Non-linear functional on $L^2$

Let $a,b,g,h$ be real numbers. How to prove that the functional $F\colon L^2 [a,b]\to \mathbb{R}$, given by $F(u)=\int_a^b (u^2(x)-gu(x)-h)\,dx$ is continuous? Thank you
2
votes
2answers
420 views

Infinite Dimensional Hilbert Space

Let $H$ be a Hilbert space with a countable basis $B$. Does it mean that any vector $x\in H$ can be expressed as a finite linear combination of elements from $x$, or as an infinite linear combination? ...
2
votes
2answers
864 views

Why is Parseval's Equality and Bessel's Inequality Different?

Bessel's Inequality: $\sum_n |\langle x, e_n \rangle |^2 \leq \|x\|^2$ Parseval: $\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ $\sum_n |\langle x, e_n \rangle |^2 = \|x\|^2$
2
votes
2answers
194 views

elliptic pdes and associated bilinear forms for Lax-Milgram

I have a simple question on elliptic pdes, actually I can not understand clearly from definitions. Thats why I want to try think on an example. Let us have an elliptic pde $$-A \Delta ...
2
votes
3answers
425 views

$C[0,1]$ is NOT a Banach Space w.r.t $\|\cdot\|_2$

I'm trying to find a cauchy sequence in $C[0,1]$ that converges under $\|\cdot\|_2$ to a limit which isn't continuous. Any ideas?
2
votes
2answers
78 views

Banach space it isn't Hilbert space [duplicate]

How can give me two or three examples about Banach spaces which it is not Hilbert spaces with proof ( I mean why it isn't Hilbert spaces ) ?
2
votes
2answers
333 views

Physical interpretation of L1 Norm and L2 Norm

In signal analysis, students have no qualms about associating the L2 norm of a square integrable function f(t) as the energy associated with that signal. A good understanding of whether a function ...
2
votes
4answers
318 views

Square root of a Hermitian operator exists

There are a lot of questions here about square root operators, but none of them addresses the basic question of existence, and I didn't find a very beefy section in Wikipedia talking about this, so ...
2
votes
2answers
148 views

Questions about the proof of the Riesz representation theorem

Let $H$ be Hilbert space, $f:H \rightarrow \Bbb F$ linear and bounded map. I'm trying to prove that there exists only one $z_0 \in H$ such that: $ \forall_{x \in H} : f(x)=\langle x,z_0\rangle$ ...
2
votes
2answers
239 views

What is the norm of the operator $L((x_n)) \equiv \sum_{n=1}^\infty \frac{x_n}{\sqrt{n(n+1)}}$ on $\ell_2$?

Let $(x_n) \subset \ell_2$ and let operator $L:\ell_2\to \mathbb R$ be defined by: $\displaystyle L((x_n)) := \sum_{n=1}^{\infty} \frac{x_n}{\sqrt{n(n+1)}}$. Find the norm of L.
2
votes
2answers
138 views

$f'$ is in $L^2[0,1]$

Let $f$ is absolutely continuous function on $[0,1]$, $f(0)=0$ and $f' \in L^2[0,1]$. Would you help me to prove that there is constant $c$ such that $$|f(t)| \leq c \left( \int_0^1 |f'(t)|^2 dt ...
2
votes
3answers
2k views

Question on weak convergence ( Example).

Can anybody tell me why $\sin(nx)$ converges weakly in $L^2(-\pi,\pi)$. I can't see how $\sin(nx)$ can converge? Explanation with any other example will be nice as well.
2
votes
2answers
36 views

Help needed in determining if statement is true

Let $H$ be a Hilbert space, and $(x_n)_{n=1}^\infty$ be a sequence in $H$ with $x_n\rightharpoonup x\in H$ weakly. Then $\|x_n\|\to\|x\|$. I found this in one of my textbook; my question is if ...
2
votes
1answer
113 views

What is a good definition of Hilbert space?

Motivation of my question: in my opinion, in view of the common definition, the statement "$\ell_p$ is a Hilbert space if and only if $p=2$" makes no sense because there is no inner product in the ...
2
votes
2answers
545 views

Isomorphism of Banach space

If $T:H\to B$ is isomorphism of Banach spaces and $H$ is Hilbert, must $B$ necessarily be Hilbert?
2
votes
2answers
160 views

I need to show that $K$ is compact and that $co(K)$ is bounded, but not closed.

Let $x_n$ be a sequence in a Hilbert space such that $\left\Vert x_n \right\Vert=1$ and $ \langle x_n,\ x_m \rangle =0 $, for all $n \neq m$. Let $ K= \{ x_n/ n : n \in \mathbb{N} \} \cup \{0\} $. ...
2
votes
2answers
165 views

Is $L^2(D)$ separable?

Let $D$ be a bounded connected open subset of $R^n$ and $μ$ is a finite measure on $D$, say the Lebesgue measure. Is $L_2(μ)$ separable? Is a bounded sequence $\{f_k\}$ of $L^2(μ)$ pre-compact?