Does there exist a Banach space $X$ which admits both a conditional and an unconditional Schauder Basis? If so, can one find an example in the collection of $\ell^p$ spaces?
My thoughts so far:
- I've been able to convince myself that this is not possible in $\ell^2$, but of course this is a very special case.
- The standard basis in $\ell^p$ is an example of an unconditional basis. So the second qeustion boils down to whether there is also a conditional one.
Also, given the type of question, I think I should mention right away that, no, this is not homework.
Thanks in advance.
It is known that every infinite dimensional Banach space with a basis has a conditional basis. This is a result of Pelczynski and Singer from 1964. A proof of this can be found in Topics in Banach Space Theory, Albiac and Kalton, page 235.
Specific examples for $c_0$ and $\ell_1$ are easy to find and describe:
For $c_0$, the sequence $(x_n)$ defined by $x_n=e_1+\cdots+e_n$ is a conditional basis called the summing basis. In $\ell_1$, the sequence $(x_n)$ defined by $x_1=e_1$ and $x_n=e_{n-1}-e_n$ for $n>1$ forms a conditional basis.
Examples of conditional bases for the other $\ell_p$ spaces are known but are not so easy to find or describe. The text by Singer mentioned in my comment above furnishes explicit examples, though. For an example for $p=2$, see also the Albiac and Kalton link in the first paragraph above. Lindenstrauss and Tzafriri's Classical Banach Spaces I also contains examples (in particular, they show $\ell_2$ has a conditional basis in Proposition 2.b.11).