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Remarks on Gteneralizations of 2-Inner Products

✍ Scribed by Charles Dminie; Siegfried Gähler; Albert White

Publisher
John Wiley and Sons
Year
2009
Tongue
English
Weight
456 KB
Volume
74
Category
Article
ISSN
0025-584X

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✦ Synopsis

Let L be a linear space with dim Lz-1 and (-, -I .) be a real function on 1. (a, a I b ) Z O ; (a, a I b) = 0 if and only if a and b are linearly dependent, L x L x L satisfying 2. (a, a I b ) = ( b , b I a ) , 3. (a, b Ic)=(b, a Ic), 4. (aa, b I c) =a@, b I c) for any real a, 6. (a +a', b 1 C) = (a, b 1 c) + (a', b I c).

(. , * 1 a ) is called a 2-inner product on L and (L, (-,

A conoept which is closely related to 2-inner product spaces is that of 2normed spaces. For a linear space L with dim L-1, let 11-, -11 be a real function on L x L satisfying 1. \la, bll= 0 if and only if a and b are linearly dependent, 3. Ilaa, bll= (a( [(a, bJI for any real a, 2-Ila, bll = lib, all, 4. Ila + b, Cll p Ila, cII + lib, ell.

]I-, -11 is called a 2-nomn on L and (L, 11. , -11) is a 2-nomned space ([4]). The 2-norm is a non-negative function.

The concepts of 2-inner product and 2-norm are 2-dimensional anaJogues of the concepts of inner product and norm. For a 2-inner product ( a , I .) the inequality I(a, b I c)l s (a, a I c)ll' (b, b 1 c)"~, a 2-dimensional analogue of the

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