A Banach space that cannot be made into a BIP space
Published online by Cambridge University Press: 24 October 2008
Extract
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
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 63 , Issue 4 , October 1967 , pp. 949 - 950
- Copyright
- Copyright © Cambridge Philosophical Society 1967
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