The (infinte dimensional) separable quotient problem asks whether every infinite dimensional Banach space has a separable infinite dimensional quotient. In the literature I have seen that is problem is attributed to Banach or to Mazur. Is it known which one of them posed the problem and when it was posed?
It seems the only firm fact about the origin of the problem is that it is discussed for the first time in print in 1969 in Rosenthal's On quasi-complemented subspaces of Banach spaces, without attribution to Banach or Mazur. The attribution to Mazur appears in Ferrando--Kakol--Lopez-Pellicer--Sliwa paper On the separable quotient problem for Banach spaces with the date of 1932, but no details. Leiderman-Morris-Tkachenko in On the separable quotient problem for Banach spaces write:
"The Separable Quotient Problem for Banach Spaces has its roots in the 1930s and, according to a private communication from Wladyslaw Orlicz to Jerzy Kakol, is due to Stefan Banach and Stanislaw Mazur"
I emailed Jerzy Kakol to clarify, and he replied:
"Formally this problem has been announced by Mazur. But I was told by prof. Orlicz, who was a student of Banach, that Banach knew already this problem much earlier than mentioned by Mazur."
How exactly Mazur announced it I am not sure. The story of the Lvov mathematical school, to which both Banach and Mazur belonged, is told in Duda's prize-winning Pearls from a Lost City. Much of the interaction took place in Lvov's cafés, and none of it was recorded until Banach's wife introduced the now famous Scottish notebook in 1935 (by then the meetings regularly took place in the Scottish Café). It is there that a cousin problem 153 (with a live goose for a prize), which was later connected to Schauder bases and solved by Enflo, was entered by Mazur on November 1936.
If Mazur "formally announced" the separable quotient problem in a café in 1932 there would be no record of it, and we can at best hope for an eyewitness testimony. The same goes for Orlicz's attribution to Banach. After Soviet takeover of Poland in 1939 the Lvov school was essentially snuffed out and after World War II its members scattered to the winds. It is possible that by word of mouth the problem eventually reached Rosenthal, who was not, however, aware of its source.