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".

Preliminaries and notation.- I. The class of operators of the form ?j?j ?j ? ??j.- 1. The operator ?j ?j ?j ? ??j.- 2. The spectrum of the operator ?j ?j ?j ? ??j.- 3. Completely continuous operators.- 4. The spectral representation of a completely continuous Hermitean operator.- 5. Some equalities and inequalities for the proper values of completely continuous Hermitean operators.- 6. Some ideals of operators.- 7. The ideals of completely continuous operators.- 8. Some uniformly closed left ideals of completely continuous operators.- II. The Schmidt-class.- 1. (? c) as a Hilbert space of completely continuous operators.- 2. The Schmidt-class of operators on L2.- III. The trace-class.- 1. (? c) as a Banach space of completely continuous operators.- 2. A maximum problem in (? c).- IV. The successive conjugate spaces of the space ? of all completely continuous operators.- 1. A characterization of ?* and ?**.- 2. ? is not a conjugate space.- 3. Some linear functionals on the space of all operators.- V. Norm ideals.- 1. Crossnorms and norm ideals.- 2. A maximum problem for ?n.- 3. Symmetric gauge functions on 𝔄n.- 4. The unitarily invariant crossnorms on ?n.- 5. The symmetric gauge functions on 𝔄 and the unitarily invariant crossnorms on ?.- 6. A special class of symmetric gauge functions.- 7. Norm ideals and the minimal norm ideals.- 8. The uniqueness of norm for the minimal norm ideals.- 9. An open problem.