A well-known open question is whether every countable collection of Lipschitz functions on a Banach space X with separable dual has a common point of Fr´chet differentiability. We show that the answer is positive for e some infinite-dimensional X. Previously, even for collections consisting of two functions this has been known for finite-dimensional X only (although for one function the answer is known to be affirmative in full generality). Our aims are achieved by introducing a new class of null sets in Banach spaces (called Γ-null sets), whose definition involves both the notions of category and measure, and showing.
