2
votes
1answer
25 views

Find the values of $p$ for which the given sequence converges in $l^p$ norm or weakly

We consider $E_p=(c_{00}, ||\cdot||_{p})$ (where $c_{00}$ are the sequences that are zero except in a finite number of values and $1\leq p \leq \infty$) and the sequence:$$(x_n)_{n\in ...
1
vote
1answer
33 views

Prove this inclusion: $\bigcup_{k<p}\ell^k\subsetneq\ell^p$

Let $1<p<\infty$. I have to prove that $$ \bigcup_{k<p}\ell^k\subsetneq\ell^p. $$ I am not able to find a counterexample to prove the inequality.
0
votes
1answer
40 views

$\ell^p$ spaces' inclusion [on hold]

$$ \ell^s\subsetneq \bigcup_{k<p}\ell^k\subsetneq \ell^p\subsetneq\bigcap_{k>p}\ell^k\subseteq \ell^q $$ for any $1\le s<p<q$. Any idea to prove these inclusions? Counterexamples for the ...
2
votes
1answer
46 views

Increasing sequence of closed subspaces of $L^2$

Not sure how to start on this question. Writing out a decomposition didnt seem to lead to anything. I think you might have to guess the answer the question first before starting a proof. Please ...
0
votes
2answers
14 views

Sequence of continuous functions with unitary norm

I want to construct a sequence $f_n$ of continuous functions in $[0,1]$ such that $||f_n||=1$ (so a bounded sequence) and $||f_n-f_m||=1$ (it doesn't have any convergent subsequences). The norm is ...
2
votes
1answer
62 views

the principle of uniform boundedness and $l^p$ space

If $1<p<\infty$ and $\{x_n\}\subset l^p$, then $\sum_{j=1}^\infty x_n(j)y(j)\to 0$ for every $y\in l^q$, $\frac{1}{p}+\frac{1}{q}=1$, iff $\sup_n||x_n||_P<\infty$ and $x_n(j)\to 0$ for every ...
2
votes
1answer
46 views

$(\ell_p,\|.\|_{\infty})$ Banach or separable

Is $(\ell_p,\|.\|_{\infty})$ for $1\leq p<\infty$ a Banach or separable space? is there any fast way of proving it without checking separability or completeness with the usual way?
5
votes
2answers
61 views

Do non-$\ell^2$ sequences have an $\ell^2$ functional that takes them to infinity?

Suppose $\{a_n\}_{n=1}^{\infty}$ is a sequence of real numbers (suppose also positive for simplicity) so that $$\sum_{n=0}^{\infty} a_n^2 = \infty$$ i.e. the sum diverges. Can you necessarily find a ...
2
votes
2answers
61 views

$\int_{\mathbb R} |f(x)| dx < \infty \implies \sum_{n\in \mathbb Z} |f(n)| < \infty $?

Put, $C_{0} (\mathbb R)=\{f:\mathbb R \to \mathbb C: f \text { is continuous on} \ \mathbb R \ \text {and } \lim_{|x|\to \pm \infty}f(x)=0 \}$(= Continuous functions on $\mathbb R$ vanishing at ...
0
votes
1answer
70 views

Constructing Sequences in Lp

Consider Banach Space $L^{p}(U)$ where $U$ is bounded open subset in $\mathbb{R}^{n}$. Take a bounded sequence $\{u_{m}\}_{m}^{\infty}$ in $L^{p}(U)$. Consider a subsequence ...
9
votes
3answers
235 views

Properties of $\bigcap_{p > 1} \ell_p$

Consider the following space of sequences $$\left\{a=(a_n)_{n\in\mathbb{N}}:a\in\bigcap_{p>1}\ell_p, a_n\in\mathbb{R}\right\}$$ What are some of its properties? What is its relation to $\ell_1$ and ...
2
votes
1answer
55 views

Convergence of $\varphi_n(x):=\frac{\varphi(nx)}{n}$ in Schwartz space

I want to find all $\varphi\in\mathcal S(\mathbb R)$ for which the sequence $\varphi_n(x):=\frac{\varphi(nx)}{n}$ converges in $\mathcal S(\mathbb R)$. The first step, I have already managed to do by ...
1
vote
1answer
520 views

Doubt in the proof that $l^{p}$ is complete

I was looking at the proof that $l^{p}$ is complete with respect to the standard metric. Suppose $x^{(n)}$ is a Cauchy sequence in $l^{p}$. Then Given $\epsilon > 0$, $\exists\,\, n_{0} \in ...
1
vote
1answer
623 views

A bounded sequence in $L^\infty$ has a weak-$^*$ convergent subsequence

Suppose $u_n$ is bounded in $L^\infty(\Omega)$. $\|u_n\|_{L^\infty(\Omega)}<M$. Then $u_{n_k}\to u$ weak star in $L^\infty(\Omega)$ for some $n_k\uparrow \infty$ and $u\in L^\infty$. I want to ...
1
vote
1answer
63 views

Equality of expressions containing supremum of some double sums

I am doing a project related to operator norm and some double sequences. In the course of proving some results, I encounter the following expressions: $\displaystyle \|\alpha\| := \sup_{\|x\|_p=1} ...
1
vote
1answer
26 views

Remainder of a series converges uniformly?

Let $B \subset \Bbb R^{\Bbb N}$ and $p \geq 1$. Suppose $$ \sup_{u\in B}\sum_{n=0}^\infty |u_n|^p \leq 1,\qquad \sup_{u\in B}\sum_{n=0}^\infty |u_{n+1}-u_n|^p\leq 1 $$ Is it true that $$ \sup_{u\in ...
5
votes
1answer
159 views

Is $(l^1 ,\|.\|)$ a Banach space?

Suppose $x=\{x_n\}\in l^1$ and $\|x\|=\sup|\sum_{k=1}^{n}x_k|$, let $\|x\|_1=\sum_{n=1}^{\infty}|x_n|$ is a norm for $l^1$ . Is $(l^1 ,\|.\|)$ a Banach space?
4
votes
2answers
268 views

Completeness proof of $\ell^p$

Say $\{x_n\}$ is Cauchy in $\ell^p$ and $x$ is its pointwise limit. To argue that $x \in \ell^p$ would the following be correct: Let $\varepsilon > 0$ and let $N$ be s.t. $n,m > N$ ...
2
votes
1answer
2k views

The $ l^{\infty} $-norm is equal to the limit of the $ l^{p} $-norms. [duplicate]

If we are in a sequence space, then the $ l^{p} $-norm of the sequence $ \mathbf{x} = (x_{i})_{i \in \mathbb{N}} $ is $ \displaystyle \left( \sum_{i=1}^{\infty} |x_{i}|^{p} \right)^{1/p} $. The $ ...
5
votes
2answers
195 views

Closed set in $\ell^1$

Show that the set $$ B = \left\lbrace(x_n) \in \ell^1 : \sum_{n\geq 1} n|x_n|\leq 1\right\rbrace$$ is compact in $\ell^1$. Hint: You can use without proof the diagonalization process to ...
1
vote
1answer
58 views

Can I conclude that $F\in\ell^1$?

Let $\ell^\infty$ be the Banach space of real bounded sequences with its usual norm and $S\subset\ell^\infty$ be the space of convergent sequences. Define $f:S\rightarrow\mathbb{R}$ by ...
1
vote
1answer
50 views

Unit vector basis in $\ell_1$

Can someone illuminate me with a hint about why it is the case that no subsequence of the unit vector basis $(e_n)$ of $\ell_1$ is weak Cauchy?
3
votes
1answer
138 views

Closure of $l_1$ in $l_\infty$

Suppose we have a set $A$ which is the set of all sequences that satisfy $|x_n|\xrightarrow{} 0$. If we consider $l_1$ to be a subset of $l_\infty$. Show that the closure of $l_1$ in $l_\infty$ equals ...