Some properties of the \(\ell _ 2\)-valued long James Banach space \(J(\eta,\ell _ 2)\) (Q1093850)

The main result of this paper is to show that the bidual \(J(\eta,\ell_ 2)^{**}\) of the long James type \(\ell_ 2\)-valued Banach space \(J(\eta,\ell_ 2)\) can be identified with transfinite matrices of scalars \([(b_{\alpha,i})\), \(i\in [0,\omega)]_{\alpha \in [0,\eta)}\) with some conditions and the norm of the element \(x^{**}\) in \(J(\eta,\ell_ 2)^{**}\) equals \(\sup_{\gamma \in [0,\eta]}\| \sum_{\alpha \in [0,\gamma)}\sum_{i\in [0,\omega)}b_{\alpha,i}\phi_{\alpha,i}\|_{J(\eta,\ell_ 2)^{**}}.\)