In the 1970s, Boris Tsirelson constructed a Banach space
which was the first one known that did not contain a copy
of one of the classical spaces c_{0} or l_{p}.
He used a clever inductive technique which has subsequently
been at the heart of many further constructions of Banach
spaces with strange properties.

Interestingly, nobody has found a way of obtaining Banach
spaces with these properties * without * using Tsirelson's
inductive approach, which leads to rather indirect definitions.
In particular, all the usual Banach spaces that come up in
nature, and have more straightforward descriptions, have
either c_{0} or l_{p} as a subspace. This
raises the following metamathematical challenge: define a
notion of `explicit definition' which encompasses all the
usual definitions of down-to-earth Banach spaces - c_{0},
l_{p}, L_{p}, C[0,1], Orlicz spaces, Lorentz
spaces, Sobolev spaces and so on - and prove that every
explicitly defined Banach space contains c_{0} or
l_{p}. Neither part of this programme has been done.