not a Schauder basis for C([0, 2π]). Bases for spaces of operators The space K(ℓ2) of compact operators on the Hilbert space ℓ2 has a Schauder basis. For every x, y in ℓ2, let denote the rank one operator . If is the standard orthonormal basis of ℓ2, a basis for K(ℓ2) is given by the sequence For every n, the sequence consisting of the n2 first vectors in this basis is a suitable ordering of the family {ej ⊗ ek}, for . The preceding result can be generalized: a Banach space X with a basis has the approximation property, so the space K(X) of compact operators on X is isometrically isomorphic to the injective tensor product If X is a Banach space with a Schauder basis such that the biorthogonal functionals are a basis of the dual, that is to say, a Banach space with a shrinking basis, then the space K(X) admits a basis formed by the rank one operators , with the same ordering as before. This applies in particular to every reflexive Banach space X with a Schauder basis On the other hand, the space B(ℓ2) has no basis, since it is non-separable. Moreover, B(ℓ2) does not have the approximation property. Unconditionality A Schauder basis {bn} is unconditional if whenever the series converges, it converges unconditionally. For a Schauder basis {bn}, this is equivalent to the existence of a constant C such that for all natural numbers n, all scalar coefficients {αk} and all signs . Unconditionality is an important property since it allows one to forget about the order of summation. A Schauder basis is symmetric if it is unconditional and uniformly equivalent to all its permutations: there exists a constant C such that for every natural number n, every permutation π of the set }, all scalar coefficients {αk} and all signs {εk}, The standard bases of the sequence spaces c0 and ℓp for 1 ≤ p < ∞, as well as every orthonormal basis in a Hilbert space, are unconditional. These bases are also symmetric. The trigonometric system is not an unconditional basis in Lp, except for p = 2. The Haar system is an unconditional basis in Lp for any 1 < p < ∞. The space L1([0, 1]) has no unconditional basis. A natural question is whether every infinite-dimensional Banach space has an infinite-dimensional subspace with an unconditional basis. This was