Completely continuous operators on a Hilbert space or even on a Banach
space have received considerable attention in the last fifty years.
Their study was usually confined to special completely continuous
operators or to the discovery of properties common to all of them (for
instance, that every such operator admits a proper invariant subspace).
On the other hand, interest in spaces of completely continuous operators
is comparatively new. Some results of this type may be found implicit in
the early work of E. SCHMIDT. Other results are "generally known" and
cannot be found explicitly in print. One of the interesting and
relatively new results states that modulo the language of BANACH (that
is, up to equivalence) the space of all operators on a Hilbert space f>
is the second conjugate of the space of all completely continuous
operators on f>. The study of spaces of completely continuous operators
on a perfectly general Banach space involves many difficulties. Some
stem, for instance, from the unsolved problem whether a completely
continuous operator on a perfectly general Banach space is always
approximable in bound by operators of finite rank. The answer is
affirmative in all the special Banach spaces considered. An affirmative
answer to the above problem is the ultimate desideratum - it ould
simplify the theory considerably. A negative answer, however, would be
equally interesting (although for us not so useful), since it would
settle negatively the open "basis problem".