Banach spaces and analytic P-ideals generated by compact sets.

With families of finite subsets of ω we can associate in a natural way a Banach space (in a way in which e.g. Schreier space is defined) and an analytic P-ideal. I will present several examples of such families with Banach spaces and ideals induced by them. I will show that non-trivial ideals generated by compact families cannot be Fσ. As a corollary we obtain certain strengthening of Ptak's lemma and Mazur's lemma.