1. A Banach space E over the complex field C is said to be a Banach inner-product (BIP) space if there exists a mapping 〈.,.〉 of E × E into C satisfying:

(i) 〈x, x〉 ≥ 0 (x ∈ E) with equality only if x = 0;

(ii) 〈x, y〉 = 〈y, x〉 (x, y ∈ E);

(iii) 〈x + λy, z〉 = 〈x, z〉 + λ〈y, z〉 (x, y, z ∈ E, λ ∈ C);

(iv) 〈x, x〉 ≤ k2 ‖x‖2 (x ∈ E),

where k is a fixed positive number. Thus 〈.,.〉 is an inner product on E, which induces a norm ‖·‖1 by the relation