Chapter I Elementary PropertiesSIEGFRIED GAHLER ([I], p. 2) defines a linear 2-normed space to be a pair (L, 11 . , -11) where L is a linear space and 11 , . (1 is a real valued function defined on L such that for a, b, c E L 1) 11 a, b 11 = 0 if and only if a and b are linearly dependent, 2) II a, b I 1GAHLER ([I], p. 4) proves that /I., Definition 1.1. A sequence {xn} in a linear 2-normed space L is called a CAUCHY sequence if there exists y, z E L such that y and z are linearly independent, the lim Ij x,, -x,, y [I = 0 and the lim I/ x,, -x,, z 11 = 0 . is a non-negative function. Theorem 1.1. Let L be a linear 2-normed space. i) If {xn> is a CAUCHY sequerhce in L with respect to a and b, then {II $9, 7 a Ill { Ik,, b Ill are real CAUCHY sequences. ii) If {xn> and {y,} ure CAUCHY sequences in L with respect to a and b, and {an> is a real CAUCRY sequence, then {xrL + y, } and {ag6 xpl} are CAUCHY sequences in L.

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Let L be a linear space of dimension greater than 1 and [j 1. 11 a, b /I = 0 if and only if a and b are linearly dependent, 2. II a, b I 1 = /I b, a /I,c 11. , * [I a realvalued function on L x L which satisfies the following four conditions: 11 -, * // is called a 2-norm on L and (L, 11 * , * 11) linear 2-normed space ([S]). The 2-norm is a non-negative function. With respect t o the dimension of the norm the notion of linear 2-normed space is a 2-dimensional analog t o the notion of normed linear space. A normed linear space is called strictly convex if /I x + y I/ = 11 x 11 + [j y 11 and I/ x /I = I[ y [I = 1 imply that x = y. I n literature other equivalent conditions for the strict convexity are well known .For example ([57) a normed linear space is strictly convex if and only if /] x + y jl = /I x /I + // y 11,x, y + 0 imply that y = a x for some a > 0.It is the purpose of this work to extend the concept of strict convexity to 2-normed spaces.For non-zero vectors a, b E L, let V ( a , b ) denote the subspace of L generated by a and b. Whenever the notation V ( a , b ) is used, it will be understood that a and b are non-zero vectors. A linear 2-normed space (L, 11 * , 11) is said to be strictly conmex if 11 a + b, c /I = [I a, c 11 + 11 b, c 11, 1) a, c I/ = 11 b, c 1) = 1, and c 6 V (a, b ) imply that a = b.If c is a fixed non-zero element of L, let V (c) denote the linear subspace oT L generated by c and let L, be the quotient space L / V ( c ) . For a E L, let a, represent the equivalence class of a with respect to V ( c ) . L, is a vector space with addition given by a, + b, = (a -/-b), and scalar multiplication by u a, = ( a a),. For arbitrary a, b E L which satisfy a, = b,, the conditions

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, (-, 2pfe-mBERT 8$)aCe ([2], [3], [7]). -)) is a 2-inner product or aA conoept which is closely related to 2-inner product spaces is that of 2-normed 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 CATJCHY-BUNJAKOWSKI inequality, holds ([2], Lemma 1). Therefore, (a, b I c)=O if a and c or b and c are linearly dependent. In [2] it is shown that [la, bll= (a, a lb)"'i, a 2-norm on (L, (. , I 9 ) ) . Every 2-inner product space will be considered to be a 2-normed space with the 2-norm 110, bll= (a, a 1 b)"'.There are many occasions where the study of 2-inner product spaces and 2-normed spaces is facilitated by oonsidering the bivectors over L. Let Bi be the 1) Reeearoh of the finst and the third named author waa mpported in part by a St. Boneventure Faoulty a r c h GrantIn-Aid.