James' space


In the area of mathematics known as functional analysis, James' space is an important example in the theory of Banach spaces and commonly serves as useful counterexample to general statements concerning the structure of general Banach spaces. The space was first introduced in 1950 in a short paper by Robert C. James.
James' space serves as an example of a space that is isometrically isomorphic to its double dual, while not being reflexive. Furthermore, James' space has a basis, while having no unconditional basis.

Definition

Let denote the family of all finite increasing sequences of integers of odd length. For any sequence of real numbers and we define the quantity
James' space, denoted by J, is defined to be all elements x from c0 satisfying
, endowed with the norm.

PropertiesMorrison, T.J. ''Functional Analysis: An introduction to Banach space theory''. Wiley. (2001)