All Questions
16 questions
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 $p > 1$. ...
- 217
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 ...
- 41.8k
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.
...
- 1,591
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 ...
- 1,591
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 ...
- 744
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}^\...
- 41.8k
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 ...
- 319
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 ?
- 151
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 \...
- 5,529
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
$$\Vert v\Vert_{\ell^p_\omega} := \left(\sum_{k=1}^\infty \omega_k^p |v|_k^p\right)...
2
votes
0
answers
184
views
Properties of the optimal decomposition for the $K$-functional between $\ell_1$ and $\ell_2$
Background: For any fixed $t> 0$, the $K$-functional defines a norm on the space $\ell_1+\ell_2$:
$$
\lVert a\rVert_{K(t)} = \inf\{\lVert a'\rVert_1+ t\lVert a''\rVert_2 : a'\in\ell_1,\ a''\in\...
- 1,372
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:
...
- 433
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:
$...
- 5,529
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}) = ...
- 1
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 $A$ ...
- 101
-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 ...
- 199