A Unified Construction Yielding Precisely Hilbert and James Sequences Spaces

Following James’ approach, we shall define the Banach space J(e) for each vector e = (e1, e2, ..., ed) ∈ R d with e1 6= 0. The construction immediately implies that J(1) coincides with the Hilbert space i2 and that J(1;−1) coincides with the celebrated quasireflexive James space J . The results of this paper show that, up to an isomorphism, there are only the following two possibilities: (i) either J(e) is isomorphic to l2 ,if e1 + e2 + ...+ ed 6= 0 (ii) or J(e) is isomorphic to J . Such a dichotomy also holds for every separable Orlicz sequence space lM . 0. Introduction In infinite-dimensional analysis and topology – in Banach space theory, two sequences spaces – the Hilbert space l2 and the James space J – are certainly presented as a two principally opposite objects. In fact, the Hilbert space is the ”simplest” Banach space with a maximally nice analytical, geometrical and topological properties. On the contrary, the properties of the James space are so unusual and unexpected that J is often called a ”space of counterexamples” (see [3,5]). Let us list some of the James space properties: (a) J has the Schauder basis, but admits no isomorphic embedding into a space with unconditional Schauder basis [1,3,4]; (b) J and its second conjugate J are separable, but dim(J/χ(J)) = 1, where χ : J → J is the canonical embedding (see [1]); (c) in spite of (b), the spaces J and J are isometric with respect to an equivalent norm (see [2]); (d) J and J ⊕ J are non-isomorphic and moreover,J and B ⊕ B are non-isomorphic for an arbitrary weakly complete B (see [3, 4]); (e) on J there exists a C-function with bounded support, but there are no C-functions with bounded support (see [7]); (f) there exists an infinite-dimensional manifold modelled on J which cannot be homeomorphically embedded into J (see [4,7]); and (g) the group GL(J) of all invertible continuous operators of J onto itself is homotopically non-trivial with respect to the topology generated by operator’s norm (see [8]), but it is contractible in pointwise convergency operator topology (see [10] and the book [3] for more references). In this paper we shall define the Banach space J(e) for each vector e = (e1, e2, ..., ed) ∈ R with e1 6= 0. The construction immediately implies that J(1) = l2 and 1991 Mathematics Subject Classification. Primary: 54C60, 54C65, 41A65; Secondary: 54C55, 54C20.