Some remarks on strictly convex semi-inner-product spaces

Unni, K. R. and Puttamadaiah, C. (1985) Some remarks on strictly convex semi-inner-product spaces. Bull. Calcutta Math. Soc., 77 (5). pp. 261-265.

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Semi-inner-products have been studied by G. Lumer [Trans. Amer. Math. Soc. 100 (1961), 29–43; MR0133024 (24 #A2860)]. The semi-inner-product [x,y] behaves linearly with respect to the first variable, with little said about the second. The quantity ∥x∥=[x,x]1/2 defines a norm which gives the original topology; furthermore, any normed linear space can be made into a semi-inner-product space. A normed space is said to be strictly convex if every point of the unit space is an extreme point (i.e., the sphere contains no segments). The authors consider the following assertions about semi-inner-product spaces: (1) ∥x+y∥=∥x∥+∥y∥, where x,y≠0, implies y=λx for some real λ>0. (2) [x,y]=∥x∥⋅∥y∥, where x,y≠0, implies y=λx for some real λ>0. (3) ∥y+z∥=∥y∥ and [z,y]=0 imply that z=0. They prove that these assertions are equivalent, and that they are equivalent to the statement that the space is strictly convex. Assertion (1) is well known; (2) was stated without proof by E. Berkson [ibid. 116 (1965), 376–385; MR0187100 (32 #4554)]; (3) seems to be new. In addition they prove that ∥x+y∥=∥x∥+∥y∥ implies [x,x+y]=∥x∥∥x+y∥ and [y,x+y]=∥y∥∥x+y∥. The proofs are complete, short, transparent and very pleasing.

Item Type: Article
Subjects: Physical Sciences > Mathematics
Divisions: PG Campuses > Manasagangotri, Mysore > Mathematics
Depositing User: Kodandarama
Date Deposited: 21 May 2013 03:40
Last Modified: 21 May 2013 03:40

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