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0 votes
1 answer
162 views

Counterexample wanted: Banach space but not BK-space

What is an example of a Banach space that is not a BK-space? A normed sequence space $X$ (with projections $p_n$) is a BK Space if $X$ is Banach space and for all natural numbers $n$, $p_n(\bar{x}) = ...
7 votes
2 answers
419 views

A counterexample showing $BV_p \neq AC_p$

I am trying to work through a supposedly simple counterexample given in papers by Love and Gehring regarding a $p$-power generalization of bounded variation and absolute continuity. Let $p > 1$. ...
2 votes
1 answer
244 views

Can a weighted $\ell^p$ norm be bounded by an unweighted $\ell^q$ norm?

For any sequence $\omega\in[1, \infty)^{\mathbb{N}}$, define the weighted $\ell^p_\omega$-norm of the sequence $v$ by $$\Vert v\Vert_{\ell^p_\omega} := \left(\sum_{k=1}^\infty \omega_k^p |v|_k^p\right)...
2 votes
1 answer
259 views

Are Chebyshev polynomials a Schauder basis of $\mathrm{Lip}[-1,1]$?

It is known that every Lipschitz function $f \colon [-1,1] \to \mathbb R$ can be expressed as a series in the Chebyshev polynomials $$f = \sum_{n = 0}^\infty a_n T_n $$ which is absolutely convergent ...
6 votes
1 answer
267 views

Unconditionally convergent series in $\ell_2$ consisting of $\ell_1$-small vectors

For a function $x:\omega\to\mathbb R$ let $|x|$ denote the function $|x|:\omega\to[0,\infty)$, $|x|:n\mapsto|x(n)|$. It is well-know that a series $\sum_{n\in\omega}r_n$ of real numbers converges ...
-1 votes
1 answer
320 views

Existence of weak limit for bouded sequence $\{y_n\}$ such that for every weak limit point $\{y_n\}$ must equal $y$

Let $X$ be separable Banach space and $\{x_n\}$ be a bounded sequence, relatively weakly compact sequence in $X$. we set $y_n=\frac{1}{n}\sum_{i=1}^{n}{x_i}$, then (together with the Krein and ...
1 vote
0 answers
56 views

Monotonicity of the norms on the sequence spaces 2

This is a complement of my previous question about the sequence spaces (I'm afraid, there will be a third part). Let $\rho:[0,+\infty)^{\mathbb{N}}\to[0,+\infty] $ satisfy the following properties: $...
2 votes
1 answer
378 views

Does the norm on a sequence space have to be monotone?

Let $\rho:[0,+\infty)^{\mathbb{N}}\to[0,+\infty] $ satisfy the following properties: $\rho(\lambda u)=\lambda\rho( u)$, for every $u\in [0,+\infty)^{\mathbb{N}}$ and $\lambda\ge 0$; $\rho(u+v)\le \...
3 votes
1 answer
269 views

What is a standard name for this kind of unconditional bases in Banach spaces?

I am looking for a standard name (if it exists) for the following property of a Schauder basis $(e_i)_{i=1}^\infty$ in a Banach space $X$: $$\|\sum_{i\in F}x_ie_i\|\le\|x\|$$for any $x=\sum_{i=1}^\...
1 vote
1 answer
232 views

A double sequence in a Banach space

Let $V$ be a infinite dimensional Banach space over $\mathbb{C}$ Let $\{a_{m,n} \cdot v_{m,n}\}_{m,n \in \mathbb{N}}$ be a double sequence with $a_{m,n} \in \mathbb{C}$ and $v_{m,n} \in V$ such that: ...
4 votes
1 answer
318 views

Non-equivalence of admitting different types of bases in Banach spaces

Whenever a certain type of (Schauder) basis is defined, it is natural to ask where that type lies in the scheme of other types of bases. This involves finding counter-examples of one type of basis ...
2 votes
0 answers
184 views

Properties of the optimal decomposition for the $K$-functional between $\ell_1$ and $\ell_2$

Background: For any fixed $t> 0$, the $K$-functional defines a norm on the space $\ell_1+\ell_2$: $$ \lVert a\rVert_{K(t)} = \inf\{\lVert a'\rVert_1+ t\lVert a''\rVert_2 : a'\in\ell_1,\ a''\in\...
6 votes
0 answers
307 views

a question about Tsirelson's space

NOTE: I asked this question over at math.stackexchange.com but got no answer or comments after 3 days, probably because it's a bit specialized. Hopefully it is interesting enough to ask over here. ...
2 votes
1 answer
373 views

Is it true that $c_0(X)^* = \ell_1(X^*)$ ?

I'm trying to prove this that but I can't . Any help/reference ?
0 votes
0 answers
255 views

Convergence of a function in a metric space to its metric.

Given a metric space $(\mathbb{A},d)$ in $\mathbb{R^n}$ 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 $A$ ...
4 votes
1 answer
2k views

Characterizations of a linear subspace associated with Fourier series

Let $c_0$ be the Banach space of doubly infinite sequences $$\lbrace a_n: -\infty\lt n\lt \infty, \lim_{|n|\to \infty} a_n=0 \rbrace.$$ Let $T$ be the space of $2\pi$ periodic functions integrable ...