You need to know: Hilbert space , Banach space B, notation for norm in B, convergence in this norm, isomorphic Banach spaces, dimension of a Banach space, finite and infinite dimensional Banach spaces, subspace of a Banach space, bounded linear operator K between Banach spaces, range of K, compact set in a Banach space.
Background: A sequence of a Banach space B is called basis of B if every has a unique representation of the form for some real numbers . Two bases and of B are called equivalent if series converges in B if and only if converges. A basis of B is called symmetric if every permutation of is a basis of B equivalent to .
A Banach space B is said to have the approximation property (AP) if for every compact set K in B and for every , there is a bounded linear operator , whose range is finite-dimensional, and such that for all . We say that Banach space B has the hereditary AP (or is a HAPpy space) if all of its subspaces have the AP.
The Theorem: On 7th November 2010, William Johnson and Andrzej Szankowski submitted to the Annals of Mathematics a paper in which they proved the existence of a HAPpy Banach space, not isomorphic to the Hilbert space , which has a symmetric basis.
Short context: The first examples of HAPpy Banach spaces not isomorphic to a Hilbert space was constructed by Johnson in 1980. Later, Pisier constructed more such examples, but none of them have a symmetric basis. In fact, Johnson asked in 1980 whether there exists any HAPpy space, other than , that has a symmetric basis. The Theorem answers this question affirmatively.
Theorems of the 21st century 