James' space

Let ${\displaystyle {\mathcal {P}}}$ denote the family of all finite increasing sequences of integers of odd length. For any sequence of real numbers ${\displaystyle x=(x_{n})}$ and ${\displaystyle p=(p_{1},p_{2},\ldots ,p_{2n+1})\in {\mathcal {P}}}$ we define the quantity

${\displaystyle \|x\|_{p}:=\left(x_{p_{2n+1}}^{2}+\sum _{m=1}^{n}(x_{p_{2m-1}}-x_{p_{2m}})^{2}\right)^{1/2}.}$

James' space, denoted by J, is defined to be all elements x from c₀ satisfying ${\displaystyle \sup\{\|x\|_{p}:p\in {\mathcal {P}}\}<\infty }$, endowed with the norm ${\displaystyle \|x\|:=\sup\{\|x\|_{p}:p\in {\mathcal {P}}\}\ (x\in \mathbf {J} )}$.

• James' space is a Banach space.
• The canonical basis {e_n} is a (conditional) Schauder basis for J. Furthermore, this basis is both monotone and shrinking.
• J has no unconditional basis.
• James' space is not reflexive. Its image into its double dual under the canonical embedding has codimension one.
• James' space is however isometrically isomorphic to its double dual.
• James' space is somewhat reflexive, meaning every closed infinite-dimensional subspace contains an infinite dimensional reflexive subspace. In particular, every closed infinite-dimensional subspace contains an isomorphic copy of ℓ₂.

• Banach space
• Tsirelson space