0
votes
0answers
52 views

a completion of $c_{00}$ under some norm

Let $X$ be the completion of $c_{00}$ under some norm $\|\cdot\|_X$ satisfying $\|e_n\|_X=1$ for all $n$, where we let $(e_n)$ denote the canonical unit vectors in $c_{00}$. In Albiac/Kalton's book ...
4
votes
0answers
42 views

a question about Tsirelson's space

Background. Let $T$ denote the Figiel-Johnson construction of the Tsirelson space, that is, the completion of $c_{00}$ under the implicitly-defined norm ...
1
vote
2answers
29 views

How to show $\{f_n\}_{n=1}^\infty$ has uniformly convergent subsequence on [0,1]?

Let $\{f_n\}_{n=1}^\infty$ a sequence of second order differentiable functions on the interval [0,1]. If $\forall n\in \Bbb N$ $f_n(0)=f_n'(0)=0$ and for all $n\in \Bbb N$ and $x \in [0,1]$ , ...
0
votes
2answers
12 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
55 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
44 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?
1
vote
1answer
40 views

Does a bounded sequence have a convergent subsequence in a Banach space?

Is there any example of a bounded sequence in a Banach space that does not have a convergent subsequence?
0
votes
1answer
49 views

Compact subsets of $c_0$

Let $c_0$ be the Banach space of all sequences converging to 0, equipped with the supremum norm. How do the compact subsets of $c_0$ look like? I could imagine that $K \subset c_0$ is compact if ...
0
votes
1answer
40 views

Proving a Space of Real Valued sequences is Banach.

Theorem: A normed vector space $(V,||\circ||)$ is a banach space if and only if for every sequence $x_n$ in $V$ with the property that $\sum ||x_n||<\infty$ we have $\sum x_n < \infty$. ...
1
vote
1answer
174 views

Show that ($\ell^1$, $\|\cdot\|_1$) is complete

Show that the vector space $\ell^1 : = \{(a_n) : \sum_n|a_n| < \infty\}$ with the norm $\|(a_n)\|_1 : = \sum_n|a_n|$ where $(a_n)$ are sequences in $\mathbb C$ is complete. Thanks in advance.
1
vote
1answer
417 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 ...
3
votes
2answers
56 views

Show that the image of $T:l^{\infty}\to l^{\infty}$, $(x_n)_n \mapsto \Big(\frac{x_n}{n}\Big)_n$ is not closed in $l^{\infty}$.

Denote the set of all bounded sequences in $\mathbb{R}$ by $l^{\infty}$, endowed with the sup norm $\lVert \rVert _{\infty}$. Define a map $T:l^{\infty} \to l^{\infty}$ as follows: $$(x_n)_n \mapsto ...
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?
1
vote
1answer
85 views

Proving $\ell_\infty$ is complete

I start by taking a Cauchy sequence $(a_i)$ in $\ell_\infty$. I denote the terms of $(a_i)$ as $f_1, f_2, f_3, \dots$ and so on. For each $x \in \mathbb{N}$, the sequence $(f_1(x), f_2(x), f_3(x), ...
3
votes
1answer
51 views

existence of invertible operator mapping one sequence pointwise to a 'nearby' sequence

Let $X$ be a Banach space and $(x_n)$, $(y_n)$, $(f_n)$ be bounded sequences in $X$, $X$, $X^*$ respectively such that $f_m(x_n)=\delta_{mn}$ $\forall m,n$ and $\epsilon=\Sigma\|x_n-y_n\|<\infty$. ...
1
vote
1answer
86 views

Dual spaces of complex sequences, show the second member is in the dual space

I'm having trouble with some of (ok, most of) the exercises in my 1st-year-master's functional analysis class, so here's one of them, hoping someone can help me out: If a sequence $(b_n)$ is ...
4
votes
2answers
245 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. $n,m > N$ ...
0
votes
1answer
1k views

About Banach Spaces And Absolute Convergence Of Seires

How to prove the following two assertions: If in a normed space $X$, absolute convergence of any series always implies convergence of that series, then $X$ is a Banach space. In a Banach space, ...
2
votes
0answers
142 views

Convergence of a function in a metric space to its metric

Given a metric space $(\mathbb{A},d)$ with a metric $d$ being the Euclidean metric, if $\lim_{t \rightarrow \infty}||A_{t+1}-A_t||\rightarrow 0$ is a convergent sequence where $A$ is a matrix with the ...
1
vote
1answer
65 views

Something weaker than the Riesz basis

I have some function $f$, real valued and continuous. I formed functions $\{f_{m,k}, k \in \mathbb{Z}, m>0\}$ such that that $\mathrm{span}\{f_{m,k}, k \in \mathbb{Z}, m>0\}$ is dense in ...
2
votes
2answers
130 views

Is $C^1(A)$ a Banach space?

Let $A \subset \mathbb R$ and consider the space $C^1(A)$. I am asked to prove that $( C^1(A), \Vert \cdot \Vert_{C^1(A)})$ is a Banach space, where $$ \Vert f(x) \Vert_{C^1(A)} = \sup_{x \in A} ...
3
votes
1answer
113 views

Space of $\mathbb{R}$ valued sequences converging to $0$. Some basic results.

Let $C_0(\mathbb{R})$ be the space of $\mathbb{R}$ valued sequences converging to $0$. Let $l_n$ be a positive sequence in $\mathbb{R}$ such that $\sum\limits_{n=1}^\infty l_n=1$. We define $$ ...
1
vote
1answer
93 views

Characterization of strong minimums with slices.

I am doing a proof of a Lemma that isn't in a book. Let $X$ a Banach space and $\emptyset\not=S\subset X$ closed of $X$. Let $f$ be a lower semicontinuous function bounded below in $S$. I have that ...
5
votes
4answers
182 views

Is there a null sequence that is in no $\ell_p$ with $p<\infty$?

Is $\bigcup_{p<\infty}\ell_p=c_0$? At least one inclusion obvious: every p-summable sequence converges to zero.
1
vote
1answer
241 views

Cauchy, Bolzano-Weierstrass, Convergence

Why is it that if for every bounded sequence we can find a convergent subsequence (in a normed vector space) then every Cauchy sequence converges (in this normed space)? Thanks.
1
vote
2answers
156 views

Sequences in $\ell_p$ spaces

Does there exist a sequence $(x_n)$ belonging to $\ell_1\cap\ell_2$ which converges in one but not the other? $(x_n)$ is of course a sequence in these spaces, so it's a sequence of sequences.
3
votes
1answer
2k views

Inequality between $\ell^p$-norms

Suppose that a sequence $x=(x_n)$ belongs both to $\ell^p$ and $\ell^q$ ($p,q>1$, $p\neq q$). Is there any inequality between $\|x\|_p$ and $\|x\|_q$. Can one $\ell^p$ be continuously embedded into ...
4
votes
2answers
327 views

Equivalent definitions of unconditional convergence

I am across two definitions for unconditional convergence for which it is not immediately obvious to me that they are equivalent. Here are the definitions. Throughout, $\frak{X}$ will denote a Banach ...
3
votes
2answers
116 views

Geometric Sums in Banach Algebra

Let $E$ be a Banach Algebra with identity, and $v\in E$, so that $||v|| < 1$. The geometric series $w = \sum_{k=0}^\infty v^k$ converges in the norm. I can show that $||w|| \le ...