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Riesz Representation Theorem on Generalized N-Inner Product Spaces Int. Journal of Math. Analysis, Vol. 7, 2013, no. 18, 873 - 882 HIKARI Ltd, www.m-hikari.com Riesz Representation Theorem on Generalized n-Inner Product Spaces Pudji Astuti Faculty of Mathematics and Natural Sciences Institut Teknologi Bandung Jln. Ganesha 10 Bandung 40132, Indonesia [email protected] Copyright c 2013 Pudji Astuti. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we develop a generalization of Riesz representation theorem on generalized n-inner product spaces. We show that any al- ternating n-linear functional on a finitely generated standard general- ized n-inner product space can be represented by a finite number of n-tuples of vectors. The proof of the theorem utilizes an inner product on a quotient tensor product space induced by the generalized n-inner product. Mathematics Subject Classification: 47A07, 46C05, 15A69 Keywords: Riesz representation theorem, inner products, generalized n- inner products, n-linear functional, alternating, tensor products 1 Introduction This paper deals with a generalization of Riesz representation theorem on gen- eralized n-inner product spaces. The concept of generalized n-inner product was introduced by Trenˇcevsky and Malˇceski [8] as a generalization of an n- inner product by Misiak [5]. Following this introduction, a number of results concerning the structure and properties of generalized n-inner product spaces can be found in literature (see [1], [2] and [6]). Gozali etal. [3] exposed a ques- tion whether Riesz type representation theorem holds on standard generalized 874 P. Astuti n-inner product spaces; that is whether any bounded n-linear functional on a separable Hilbert space can be represented by n unique vectors. We show that the question has negative answer and prove an enhancement of Riesz representation theorem on standard generalized n-inner product spaces. In this paper we proceed as follows. In section 2 we review the concept of generalized n-inner product and n-linear functional, and also present two examples which lead to the main result. The main result of this paper is Theorem 2.6 where the proof will be presented in section 4. Section 3 will discuss auxiliary results concerning induced inner product on a quotient tensor product space which will be used later. 2 Definitions, notation and main theorem The notation R stands for the real number field and n is a positive integer. The concept of a generalized n-inner product was introduced by Trenˇcevsky and Malˇceski [8] as a generalization of an n-inner product by Misiak [5] which, to some extent, is more proper notion as a generalization of an inner product. Definition 2.1. [8] Let V be a real vector space having dimension at least n. A real valued function with domain V 2n, the Cartesian product of 2n times of V , −,...,−|−,...,− : V n × V n −→ R is called generalized n-inner product on V if it satisfies the following conditions. i) (Quasi Positive Definite) a1,...,an|a1,...,an≥0 for all a1,...,an ∈ V and a1,...,an|a1,...,an =0if and only if the set {a1,...,an}⊆V is linearly dependent. ii) (Quasi Symmetry) a1,...,an|b1,...,bn = b1,...,bn|a1,...,an for all a1,...,an, b1,...,bn ∈ V . iii) (Linear at the first variable) λa1,...,an|b1,...,bn = λa1,...,an|b1,...,bn a1+c,...,an|b1,...,bn = a1,...,an|b1,...,bn+c, a2,...,an|b1,...,bn for all λ ∈ R and a1,...,an, b1,...,bn, c ∈ V . iv) (Quasi Skew-Symmetry) a1,...,an|b1,...,bn = −aσ(1),...,aσ(n)|b1,...,bn for all a1,...,an, b1,...,bn ∈ V and σ odd permutation on {1,...,n}. Riesz representation theorem 875 v. If a1, b1,...,b(i−1), b(i+1),...,an|b1,...,bn =0for all i ∈{1, 2,...,n} then for any a2,...,an ∈ V a1, a2,...,an|b1,...,bn =0. A vector space V coupled with a generalized n-inner product on it is called a generalized n-inner product space or n-prehilbert space. A standard generalized n-inner product space is an inner product space coupled with the standard generalized n-inner product induced by the inner product [8]. Let V be an inner product space where its inner product is denoted by −, −. The standard generalized n-inner product on V induced by the inner product −, −, is the function n n −,...,−|−,...,−S : V × V −→ R a1, b1a1, b2 ... a1, b n a2, b1a2, b2 ... a2, b | n a1,...,an b1,...,bn S = . .. . an, b1an, b2 ... an, bn for all a1,...,an, b1,...,bn ∈ V . The proof of −,...,−|−,...,−S being a generalized n-inner product can be found in [8]. In this paper we deal with standard generalized n-inner product spaces. One interesting research topic connected to generalized n-inner product spaces is multilinear function. Definition 2.2. [7] Let V,W be two real vector spaces. A real function f : V n −→ W is called multilinear or n-linear function if it is linear in each variables. That is, if f(u1,...,uk−1,αv1 + βv2, uk+1,...,un)= αf(u1,...,uk−1, v1, uk+1,...,un)+βf(u1,...,uk−1, v2, uk+1,...,un) for all k ∈{1,...,n}, α, β ∈ R, v1, v2 ∈ and ui ∈ V with i = k. A multilinear function f with codomain W = R the real number field is called multilinear or n-linear functional. 876 P. Astuti Let V be an n-prehilbert space with the generalized n-inner product de- noted by −,...,−|−,...,− and a1,...,an ∈ V . For these n vectors, one can define a functional n −→ f(a1,...,an) : V R n by mapping for any (x1,...,xn) ∈ V to | f(a1,...,an)(x1,...,xn)= x1,...,xn a1,...,an The property that −,...,−|−,...,− being multilinear induces multilinear property of the functional f(a1,...,an). The converse of this result, exposed by [3] as Riesz type representation theorem, is an interesting problem to be ad- dressed. Unfortunately the expectation that any bounded n-linear functional on a real separable Hilbert space can be represented by n unique vectors is not true as shown in the following example. Example 2.3. Let V be a standard generalized 2-inner product space with the inner product denoted by −, −. The standard generalized 2-inner product on V ; is defined as u1, v1u1, v2 u1, u2|v1, v2S = u2, v1u2, v2 for all u1, u2, v1, v2 ∈ V . For this space, the inner product −, − is 2-linear functional but it does not satisfy condition that there exist b1, b2 ∈ V such that x1, x2 = x1, x2|b1, b2S for all x1, x2 ∈ V. (1) Suppose there do exist b1, b2 ∈ V such that (1) holds. Then we have for v ∈ V v =0 , v, v = v, v|b1, b2S =0, which is a contradiction to the positive definite property of the inner product. Definition 2.4. [7] A multilinear function f : V n −→ W is called alter- nating if it satisfies f(u1,...,ui,...,uj,...,un)=0 for all u1,...,u ∈ V with u = u for some i<j. n i j Further investigation concerning Riesz type representation problem ob- serves that the n-linear functional on an n-prehilbert space generated by n vectors is alternating. This gives an idea to confine the investigation of Riesz type representation problem in the subspace of bounded alternating n-linear functionals. However, similar to the above observation, a negative answer to this inquiry is obtained as shown in the following example. Riesz representation theorem 877 Example 2.5. Let R4 be the Euclidean space coupled with the standard gen- 4 eralized 2-inner product, for every u1, u2, v1, v2 ∈ R u1, v1u1, v2 u1, u2|v1, v2 = u2, v1u2, v2 4 4 Let {e1, e2, e3, e4}⊆R be the standard basis of R . Consider the 2-linear 4 4 functional defined on R as follows; for every x1, x2 ∈ R f(x1,,x2)=x1, x2|e1, e2 + x1, x2|e3, e4 The 2-linear functional f is alternating. Suppose f can be represented by two 4 4 vectors in R , that is there exist b1, b2 ∈ R satisfying 4 f(x1, x2)=x1, x2|b1, b2 for all x1, x2 ∈ R . (2) t t Let b1 =(b11,b12,b13,b14) , b2 =(b21,b22,b23,b24) . Then we have the following equations f(e1, e2)=1=b11b22 − b21b12 f(e1, e3)=0=b11b23 − b21b13 f(e1, e4)=0=b11b24 − b21b14 f(e3, e4)=1=b13b24 − b23b14 The first three equations imply that (b11,b21) =(0 , 0) (b13,b23)=α(b11,b21) (b14,b24)=β(b11,b21) for some α, β ∈ R. As a result we obtain {(b13,b23), (b14,b24)} is linearly dependent. In contrast the last equation implies that {(b13,b23), (b14,b24)} is linearly independent, which is a contradiction statement to the previous result. Thus f cannot be represented by two vectors in R4. Even though, to a certain extent, the above Example 2.3 and Example 2.5 are disappointed, the expression of the 2-linear functional f as sum of two gen- eralized 2-inner products inspires to investigate further: can any alternating n-linear functional be represented as finite sum of generalized n-inner prod- ucts? The following theorem is the main result of this paper and it can be recognized as an enhancement of Riesz representation theorem on standard n-prehilbert spaces. Theorem 2.6. Let V be a finitely dimensional standard generalized n-inner product space with the n-inner product −,...,−|−,...,−S is induced by an − − ≥ n −→ R inner product , on V and DimÊ (V )=m n.Letf : V be an 878 P.
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