Analytic P-ideals and Banach spaces

I will talk about certain symmetries between analytic P-ideals and Banach spaces with unconditional bases (e.g. I will give some examples of Banach spaces motivated by ideals and vice-versa). I will present a theorem saying that ideals induced by compact families of finite sets are not F_sigma (and I will show that one can strengthen Mazur's Lemma using this fact).
This is a joint work with Barnabas Farkas.