A new 2-inner product on the space of p-summable sequences

In this paper, we wish to define a 2-inner product, non-standard, possibly with weights, on ℓp. For this purpose, we aim to obtain a different 2-norm ‖., .‖2,v,w, which is not equivalent to the usual 2-norm ‖., .‖p on ℓp (except with the condition p=2p=2), satisfying the parallelogram law. We discuss the properties of the induced 2-norm ‖., .‖2,v,w and its relationships with the usual 2-norm on ℓp. We also find that the 2-inner product ⟨., .|.⟩v,w is actually defined on a larger space.