All Questions
Tagged with lp-spaces sequences-and-series
210
questions
1
vote
1
answer
34
views
Finding the conjugate operator of the following operator
Let $A$ be linear operator from $l^2$ to $l^2$ such that $Ax =y^0 \cdot \sum_{1}^{\infty}{x_k} $ , where $y^0 \in l^2$ — fixed element.
Show, that conjugate operator $A^*$ exists and find it. Show, ...
- 21
0
votes
1
answer
70
views
Move Infinite sum inside a limit $t \to \infty$.
This may be simple, but I want to know if my reasoning is ok. I came across a problem whose essential set up is:
let $f_k$ be a sequence of functions in $L^1(\mathbb{R})$ (Lebesgue integrable ...
- 495
0
votes
1
answer
23
views
Sequence of sequences $\{a^{(n)}\}_n \subseteq \ell^2$ with bounded members $|a^{(n)}_k| \leq 1$ has converging subseq.
I struggling to understand a partial step in the solution to an exercise:
Given a seq. of seq. $\{a^{(n)}\}_{n \in \mathbb N} \subseteq \ell^2$ such
that $|a^{(n)}_k| \leq 1 \forall n,k \in \mathbb N$...
- 139
1
vote
0
answers
34
views
Sequence in $\ell_p$ spaces
The sequence given by:
$$x_n=(1^{-1/q},2^{-1/q}-1^{-1/q}, 3^{-1/q}-2^{-1/q}, \dots)$$
That is,
$$\sum_{n=1}^{\infty}n^{-1/q}-(n-1)^{-1/q}$$
Is this sequence in the sequence space $\ell_p$ ? where for $...
- 115
3
votes
1
answer
51
views
Can the unit ball in $\ell^2$ be a countable union of rectangle set $\prod_{n\in\mathbb{N}}[a_n,b_n)$?
I know that the unit ball in $\mathbb{R}^n$ can be a countable union of rectangles $[a_1,b_1)\times\cdots\times[a_n,b_n)$.
The question I have is whether the unit ball in infinite dimensional space, e....
- 43
1
vote
1
answer
95
views
Brezis' exercise 8.28.11: how to prove $\sum_{k=0}^{\infty} \left | \frac{1}{2} \alpha_k (f) - a \right |^2 < + \infty$?
Let $I$ be the open interval $(0, 1)$ and $H := L^2 (I)$ equipped with the usual inner product $\langle \cdot, \cdot \rangle$. Consider the linear map $T: H \to H$ defined by
$$
(Tf) (x) = \int_0^x t ...
- 17.6k
1
vote
0
answers
111
views
A question about probability theory and a little functional analysis
Consider the sequences of mean zero random variables $(X_n)_{n \in \mathbb N}$ with finite variances, i.e., $E[X_n]=0$ and $E\left[X_n^2\right] =: \sigma_n< \infty$, for each $n$. Moreover, ...
- 75
1
vote
1
answer
80
views
A question about the $||\cdot ||_{\infty}$ and $||\cdot ||_{2}$ norms in $\ell^2$
I'm a little curious to understand the $||\cdot ||_{\infty}$ and $||\cdot ||_{2}$ norms a little better.
The following example presents a sequence $(x_n)_{n \in \mathbb N}$ in $\ell^2$ that is in the ...
- 75
1
vote
2
answers
50
views
Proving a subset of $l^2$ is not closed
We consider $\ell^2$ equiped with its usual inner product and let $ u = (1/2, 1/4, 1/8 , \dots , 1/2^n, \dots) \in \ell^2 $ and $$F:= \{ x = (x_1,x_2, \dots ) \in \ell^2 : \langle u,x \rangle = 0, \...
- 13
0
votes
1
answer
63
views
Proof by contradiction that $(\ell^p)^* \subseteq \ell^q$
For $p \in (1, \infty)$, let $\ell^p = \{(x_n) : \sum |x_n|^p < \infty\}$ and let $(\ell^p)^*$ denote the dual space of bounded linear forms on $\ell^p$ $(\mathbb{F} = \mathbb{R})$.
It is ...
- 2,466
0
votes
1
answer
126
views
Example of a sequence in the closed unit ball of $\ell^1$ which does not have any weakly convergent subsequence.
I am looking forward to having an example of a sequence in the closed unit ball of $\ell^1$ which does not have any weakly convergent subsequence. I need this to prove that the closed unit ball in $\...
- 835
1
vote
1
answer
81
views
Convergence of sums in $\ell^p \implies \ell^{p-\epsilon}$
Supose $\displaystyle(b_n)_{n \in \mathbb{N}}$
is a sequence of positive real numbers that
$$\displaystyle\sum_{n \in \mathbb{N}}(b_n)^{2} <\infty.$$
Does exists some $\epsilon>0$ such that $\...
- 33
1
vote
1
answer
73
views
Prove that a certain subspace is weak-*dense in $l^{1}.$
I am self-studying the Rudin's book about functional analysis; I am currently stuck on a detail of exercise 10, chapter 3, point c.
Let $\ell^1$ be the space of all real functions $x(m,n)$ on $\Bbb N\...
2
votes
1
answer
45
views
Norm of addition linear Operator
I want to find the operator norm of
\begin{equation}
Ax=(x_1+x_2,x_2+x_3,...)
\end{equation}
for x$\epsilon \ell_2$
I can show that this operator is bounded, as
\begin{equation}
\sum_{i=0}^\infty|x_i+...
- 61
1
vote
1
answer
243
views
Compact embedding $L^p$ to $L^q$
Prove that this embedding is not compact $\ell^p\subset \ell^q\subset c_0$ , where $1 \leq p < q < \infty$
I understand that this embedding is not compact, but I cannot construct an example ...
- 121