BIORTHOGONALITY IN THE REAL SEQUENCE-SPACES L(P)

In this paper we generalise some of the results obtained in [1] for th e n-dimensional real spaces l(p)(n) to the infinite dimensional real s paces l(p). Let p > 1 with p not equal 2, and let x be a non-zero real sequence in l(p). Let epsilon(x) denote the closed linear subspace sp anned by the set {x}(+/-) of all those sequences in l(p) which are bio rthogonal to x with respect to the unique semi-inner-product on l(p) c onsistent with the norm on l(p). In this paper we show that codim epsi lon(x) = 1 unless either x has exactly two non-zero coordinates which are equal in modulus, or x has exactly three non-zero coordinates alph a,beta,gamma with \alpha\ greater than or equal to \beta\ greater than or equal to \gamma\ and \alpha\(p) > \beta\(p) + \gamma\(p) In these exceptional cases codim epsilon(x) = 2. We show that {x}(+/-) is a lin ear subspace if, and only if, x has either at most two non-zero coordi nates or x has exactly three non-zero coordinates which satisfy the in equalities stated above.