3
votes
1answer
16 views

Subspace of certain series in a Hilbert space is compact

Let $E$ be a Hilbert space and let $\{x_{n}\}$ be an orthonormal basis.  Let $\{c_{n}\}$ be a sequence of positive numbers such that $\sum c_{n}^{2}$ converges.  Let $C$ be the subset of $E$ ...
4
votes
2answers
51 views

$\left\{x\in H: 2\leq \|x\|\leq 5\right\}$ is compact?

In a Hilbert space $H$ of dimention infinite, $A=\left\{x\in H:2\leq \|x\|\leq 5\right\}$ is compact? (totally bounded and complete) Thanks in advance.
1
vote
1answer
289 views

In a Hilbert space, every bounded and closed set is weakly relatively compact.

My aim is to prove that in a Hilbert space, any sequence has a weakly convergent subsequence. To prove this, I'm trying to prove that: ...
2
votes
1answer
67 views

Hilbert space, orthonormal system, compact set of vectors

Could you help me solve this problem? Let $e_1, e_2, ...$ be an orthonormal system in a Hilbert space, $\delta_1, \delta_2 ... \in (0, + \infty)$. Prove that the set of all vectors $\sum _{n=1} ...
1
vote
1answer
62 views

Sufficient and necessary condition for compact ellipsoids in $l_2$

Another fun problem from functional analysis that I am having issues with. I have fought long enough and would like to offer this to the community. For a sequence $\mathbb{R}\ni a_i>0$ consider a ...
1
vote
1answer
96 views

$\sum c_k^2<\infty$ then $A=\{\sum_{k=1}^{\infty} a_ke_k :|a_k|\leq c_k \}$ is compact

Let $\{e_k\}_{k=1}^\infty$ be an orthonormal set in a Hilbert space $H$. If $\{c_k\}_{k=1}^\infty$ is a sequence of positive real numbers such that $\sum c_k^2<\infty$, then the set: ...
0
votes
1answer
120 views

Can a Accumulation Point be an Eigenvalue?

I have a discrete (separable) infinite dimensional Hilbert Space with a compact operator defined on it. So 0 is an accumulation point (some theorem says so). Can 0 also be an eigenvalue? And how would ...
7
votes
1answer
99 views

Tight Probability on Hilbert space

I am considering the following problem. Let $(X_j)$ be i.i.d. $N(0,1)$ random variables and $H$ a Hilbert space with orthonormal basis $(e_j)$. Let $$X:=\sum_j \frac{X_j e_j}{j}$$ And for any ...
5
votes
1answer
155 views

Show $T$ is compact

$H$ and $K$ are Hilbert Spaces, $(u_n)$ and $(v_n)$ are sequences in $H$ and $K$ respectively. $\sum_{n=1}^{n=\infty} \|u_n\|\|v_n\| $ converges. $T\colon H\rightarrow K$ is defined by ...
4
votes
1answer
110 views

Is it possible to 'approximate' compact, convex sets in $\ell^2$ by the Hilbert cube

Define $H=\{(x_n)_n\in\ell^2:|x_n|\le \frac1n, n\in\mathbf N\}\subset\ell^2$. This set is known as the Hilbert cube and it is well-known that $H$ is compact, convex and non-empty. Let ...
8
votes
5answers
858 views

How to show that this set is compact in $\ell^2$

Let $(a_n)_{n}\in\ell^2:=\ell^2(\mathbb{R})$ be a fixed sequence. Consider the subspace $$C=\{(x_n)_{n}\in\ell^2 : |x_n|\le a_n\text{ for all }n\in\mathbb{N}\}.$$ According to the book [Dunford and ...
-1
votes
1answer
141 views

Stone-Cech compactification of the separable Hilbert space

Where can I read about the Stone-Cech compactification of the separable Hilbert space?
2
votes
2answers
149 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\} $. ...