Jerzy KAKOL

On the separable quotient problem


Abstract: One of the famous unsolved problems of functional analysis asks 
(Mazur's problem (1932)) if every (infinite-dimensional) Banach space E has 
an (infinite-dimensional) separable quotient. Many Banach spaces are known 
to have separable quotient, for example, reflexive Banach spaces, or even 
weakly compactly generated Banach spaces. Rosenthal (independently Lacey) 
proved that any Banach space C(X) has separable quotient.  A general result 
of Saxon and Wilansky states that a Banach space E has separable quotient 
iff E contains a dense non-barrelled subspace. Later on, Saxon and Narayanaswami 
showed that every (LF)-space possesses separable quotient. In the first part 
we gather a few other problems from Banach space theory which are equivalent 
to the separable quotient problem and  provide recent results due to Kakol, 
Saxon and Todd concerning the same problem but posed for spaces Cp(X) and Ck(X).