Albert White

2-INNER PRODUCT SPACES II1)

This paper i s ail extension of [ l ] . We continue the i n vest igat ions begun there concerning 2-inner product spaces (2-pre-Hilbert spaces) and deal especial ly with orthogonal and orthonormal s e t s in such spaces. The reader should consult [1] for def in i t ions and notations. Let (L, (. , . I . )) be a 2-inner product space and II . , . II the associated 2-norm. L e m m a 1. For a , b , c e L ( a , b l a ) = 0 and o (a,b|yc) = f ( a , b l c ) , jr r e a l .

Some Inequalities in 2-inner Product Spaces

In this paper we extend some results on the refinement of Cauchy-Buniakowski-Schwarzs inequality and Aczels inequality in inner product spaces to 2-inner product spaces.

Heron's formula in inner product spaces

The classical Herons formula says that, for a triangle having sides of the length a, b, c and the area K, the relation Κ = [s(s o)(s b)(s c)]/ holds, where s = (a + b + c)/2. Recently, C. S. Lin proved Herons formula in 3-dimensional spaces ([7]). In this paper, we give Herons formula in inner product spaces and some properties of this formula. Also, we define a 2-inner product in terms of Herons formula. Let ( Χ , II · II) be a normed linear space. For any x,y,z G X, define (*) A(x,y,z) = [ s ( s a ) ( s b ) ( 8 c ) ] 1 / 2 , where a = ||a: b = ||a: — 2||,c = ||¡/ — z|| and s = (α + b + c)/2. From (*), algebraic calculations establish the following two theorems: T H E O R E M 1. (1) A(x,y,z) is invariant with respect to the order of the points χ, y, z, ( 2 ) A(x + u,y + u,z + u) = A(x,y, z), ( 3 ) A(px,py,pz) = pA(x,y,z) for any real number p. 1991 Mathematics Subject Classification: 46B20, 46B99.

EXTREME POINTS AND STRICT CONVEXITY

(Ni) ||a, 6|| = 0 if and only if a and b are linearly dependent, (N2) \\a,b\\ = \\b,a\\, (N3) ||aa, 6|| = |a|||a,6||, where a is real, (Nt) ||a + M | < | M | | + | M | . ||.,.|| is called a 2-norm on X and (X, ||M .||) is called a linear 2-normed space ([10]). Note that the 2-norm is non-negative and ||a, 6|| = ||a + b, 6||. The notion of a 2-semi-inner product was introduced by A. U. Siddiqui and S. M. Rizvi ([17]). But Y. Ho and A. White ([12]) modified one part of the definition of A. U. Siddiqui and S. M. Rizvi. Let [., .|.] be a real-valued function defined on X X X X X satisfying the following conditions: