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 alternating n-linear functional on a ﬁnitely generated standard generalized 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 generalized 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 question 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.

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 denoted 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 alternating if it satisfies

f(u1,...,ui,...,uj,...,un)=0 for all u1,...,u ∈ V with u = u for some i

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 generalized 2-inner products inspires to investigate further: can any alternating n-linear functional be represented as ﬁnite sum of generalized n-inner products? 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. Astuti alternating n-linear functional. Then there exists a ﬁnite number of elements n (ai1,...,ain) ∈ V ,i=1,...,k such that

k f(x1,...,xn)= x1,...,xn|ai1,...,ainS for all x1,...,xn ∈ V. =1 i

The proof of the theorem will be explained in section 4 and it utilizes properties of the induced inner product on a quotient tensor product space.

3 Induced inner product

Deﬁnition 3.1. [7] Let V be a real vector space. The n-fold tensor product of V is a real vector space T such that there exists a multilinear function, called the tensor map, t : V n −→ T having the property that any multilinear function f : V n −→ W , where W is a real vector space, can be decomposed by t as f = g ◦ t for a unique linear map g : T −→ W.

It is well known that the n-fold tensor product of V is unique up to iso- morphism. One way to construct the n-fold tensor product of V is as shown in [7]. The tensor product space can be identiﬁed by the quotient space F/S where F is the real vector space with the Cartesian product V n as its basis and S is its subspace generated by the set of all elements F of the form

α(u1,...,uk−1, v1, uk+1,...,un)+β(u1,...,uk−1, v2, uk+1,...,un)

−(u1,...,uk−1,αv1 + βv2, uk+1,...,un) for some α, β ∈ R and v1, v2, ui ∈ V with i = k. In this case, the tensor map t is deﬁned as

t(u1,...,un)=(u1,...,un)+S for all u1,...,un ∈ V . From now on, the n-fold tensor product of V will be ⊗n n denoted by V and the coset (u1,...,un)+S, for any (u1,...,un) ∈ V , called decomposable tensor, will be denoted by

u1 ⊗ u2 ···⊗un =(u1,...,un)+S

⊗n As a consequence, any element of V is a ﬁnite sum of decomposable tensors ⊗···⊗ i ui1 uin. Riesz representation theorem 879

Let V be an m-dimensional n-prehilbert space, m ≥ n, with the generalized n-inner product, denoted by −,...,−|−,...,−S, is induced by an inner product on V −, −. One can construct a bilinear form on the n-fold tensor ⊗n product space V induced by the generalized n-inner product as the following. ⊗ ⊗ ∈ ⊗n For any i ui1 ...uin, j vj1 ...vjn V define ⊗ ⊗ | i ui1 ...uin, j vj1 ...vjn t = i j ui1,...,uin vj1,...,vjn S It can be shown that the above definition is a well defined real valued function on V ⊗n × V ⊗n. Moreover, as a results of properties of the generalized n-inner product, we obtain that the functional

⊗n ⊗n −, −t : V × V −→ R is a symmetric bilinear form on V ⊗n. However, it is degenerate. We have that for any v ∈ V , v ⊗···⊗v, u1 ⊗···⊗u2t = 0 for all u1,...,un ∈ V . Whereas for v = 0 we have v ⊗···⊗v =0. According to Greub [4] the above symmetric bilinear form on V ⊗n can induce a non-degenerate symmetric bilinear form on the quotient space V ⊗n/D where D is the subspace of V ⊗n generated by all degenerate elements. That is

⊗n ⊗n D = {X ∈ V |X, Y t = 0 for all Y ∈ V }. The induced bilinear between cosets X + D, Y + D ∈ V ⊗n/D is deﬁned as

X + D, Y + Dq = X, Y t.

⊗n Now we shall show that the bilinear −, −q is an inner product on V /D. It is enough by showing that V ⊗n/D has an orthonormal basis with respect to −, −q. Let B = {b1, b2,...,bm} be an orthonormal ordered basis of V . According to Roman [7] we obtain ⊗n { ⊗···⊗ | ∈ } B = bi1 bin bi1 ,...,bin B is a basis of V ⊗n. Let us partition B⊗n becomes three subsets: ⊗n { ⊗···⊗ ∈ ⊗n | ∈ } B1 = bi1 bin B bi1 ,...,bin B,ij ik ⊗n { ⊗···⊗ ∈ ⊗n | ∈ } B3 = bi1 bin B bi1 ,...,bin B,ij = ik for some j = k .

⊗n 1. Because of the orthonormality of B, the subset B1 is orthonormal with respect to the bilinear form −, −t. ⊗· · ·⊗ ∈ ⊗n { } 2. For every bi1 bin B2 there exists σ a permutation on i1,i2,...,in ⊗···⊗ ∈ ⊗n ¯⊗n such that bσ(i1 ) bσ(in) B1 . Let B2 be the set of elements ⊗···⊗ − − sgn(σ) ⊗···⊗ bi1 bin ( 1) bσ(i1) bσ(in) 880 P. Astuti

⊗···⊗ ∈ ⊗n { } for all bi1 bin B2 where σ is the permutation on i1,i2,...,in ⊗···⊗ ∈ ⊗n ⊗n ∪ ¯⊗n ∪ such that bσ(i1) bσ(in) B1 . We obtain that the set B1 B2 ⊗n ⊗n ⊗n B3 forms a basis of V . Moreover B¯2 is a subset of D.

⊗n 3. Since any element in B3 is generated by less then n elements of B,we ⊗n obtain B3 ⊆ D.

⊗n ⊗n ⊗n ⊗n Up to now we obtain that B1 ∪ B¯2 ∪ B3 is a basis of V and the inde- ⊗n ⊗n ⊗n ⊗n pendent subset B¯2 ∪ B3 is inside D. If we can show that B¯2 ∪ B3 is a ⊗n basis of D and because B1 is orthonormal with respect to −, −t, we obtain that the set

{ ⊗···⊗ | ⊗···⊗ ∈ ⊗n} bi1 bin + D bi1 bin B1

⊗n is an orthonormal basis of V /D with respect to −, −q. ⊗n ⊗n ⊗n Now we shall show that B¯2 ∪ B3 is a basis of D. Let X ∈ V with ⊗n ⊗n ⊗n X/∈ Span{B¯2 ∪ B3 }. Write X = X1 + X2 where X1 ∈ Span{B1 } and ⊗n ⊗n X2 ∈ Span{B¯2 ∪ B3 }. Then X1 = 0 and ⊗···⊗ X1 = αi1...in bi1 bin b ⊗···⊗b ∈ ⊗n i1 in B1

⊗···⊗ ∈ ⊗n with some αi1...in = 0. Let bi1 bin B1 with αi1...in = 0. Then, ⊗n because of the orthonormality of B1 with respect to −, −t, ⊗···⊗ ⊗···⊗ X, bi1 bin t = X1, bi1 bin t = αi1...in =0

⊗n ⊗n which results in X/∈ D. Thus, we have B¯2 ∪ B3 is a generator of D which ⊗n ⊗n results in B¯2 ∪ B3 is a basis of D.

4 The proof of theorem 2.6

Let V be an m-dimensional standard generalized n-inner product space, m ≥ n, with the n-inner product −,...,−|−,...,−S induced by an inner product −, − on V . Let f; V n −→ R be an alternating n-linear functional. Referring to the above explanation let g : V ⊗n −→ R be the unique linear functional such that f = g ◦ t where t : V n −→ V ⊗n is the tensor map satisfying

t(u1,...,un)=u1 ⊗···⊗un for all u1,...,un ∈ V . We shall show that the degenerate subspace D is contained in the kernel of g so that we can lift g to the induced inner product ⊗n ⊗n ⊗n space V /D. It is enough by showing the basis B¯2 ∪ B3 ⊆ Ker(g). Riesz representation theorem 881

⊗···⊗ − − sgn(σ) ⊗···⊗ ∈ ¯⊗n Let bi1 bin ( 1) bσ(i1) bσ(in) B2 . Then

⊗···⊗ − − sgn(σ) ⊗···⊗ g(bi1 bin ( 1) bσ(i1) bσ(in)) ⊗···⊗ − − sgn(σ) ⊗···⊗ = g(bi1 bin ) ( 1) g(bσ(i1) bσ(in)) − − sgn(σ) = g(t(bi1 ,...,bin )) ( 1) g(t(bσ(i1),...,bσ(in)) − − sgn(σ) = f(bi1 ,...,bin )) ( 1) f(bσ(i1),...,bσ(in)) =0

⊗···⊗ ∈ ⊗n since f is alternating. Let bi1 bin B3 . Then bij = bik for some j = k. Hence, since f is alternating, we obtain

⊗···⊗ g(bi1 bin )=g(t(bi1 ,...,bin )=f(bi1 ,...,bin )=0.

Thus D ⊆ Ker(g). According to [7] there exists a unique linear functional

g¯ : V ⊗n/D −→ R such that g =¯g ◦ πD where

⊗n ⊗n πD : V −→ V /D

⊗n is the canonical projection deﬁned as πD(X)=X + D for all X ∈ V .Now we haveg ¯ a linear functional on V ⊗n/D, an inner product space with the inner product −, −q, Riesz representation theorem [7] guarantees the existence of a unique A + D ∈ V ⊗n/D such that

⊗n g¯(X + D)=X + D, A + Dq for any X + D ∈ V /D.

⊗n Consider that A ∈ V is of the form as a ﬁnite sum of decomposable tensors. ∈ n k ⊗···⊗ Let (ai1,...,ain) V for i =1, 2,...,k such that A = i=1 ai1 ain. Then we obtain for any x1, x2,...,xn ∈ V

f(x1,...,xn)=(g ◦ t)(x1,...,xn) = g(t(x1,...,xn)) = g(x1 ⊗···⊗xn) =(¯g ◦ πD)(x1 ⊗···⊗xn) =¯g(πD(x1 ⊗···⊗xn)) =¯g(x1 ⊗···⊗xn + D) = x1 ⊗···⊗xn + D, A + Dq ⊗···⊗ = x1 xn,At ⊗···⊗ k ⊗···⊗ = x1 xn, i=1 ai1 ain t k | = i=1 x1,...,xn ai1,...,ain S. 882 P. Astuti

5 Concluding remark

In this paper we have developed an adaptation of Riesz representation theorem for alternating multilinear functionals on standard n-prehilbert spaces. We obtained that any alternating multilinear functional on a ﬁnitely generated inner product space can be represented by a ﬁnite number of n-tuples of vectors. The uniqueness of the representation n-tuples of vectors is not guaranteed since any n-tuple of vectors which forms a degenerate decomposable tensor can be added to the representation n-tuples. However it would be interesting to study whether the minimum number of representation n-tuples of vectors is connected to the rank matrix of the associated tensor product. Extension of the result to any n-prehilbert spaces would also be an interesting problem to be investigated.

References

[1] R. Chugh and S. Lather, On Generalized n-inner product spaces, Navi Sad J. Math. 41 (2011), 73–80.

[2] R. Chugh and Sushma, Some results in Generalized n-inner product spaces, Internat. Math. Forum 4 (2009), 1013–1020.

[3] S.M. Gozali, H. Gunawan, and O. Neswan, On n-norm and bounded n- linear Functionals in a Hilbert Space, Ann. Funct. Anal. 1 (2010), 72–79.

[4] W.H. Greub, Linear Algebra, third ed., Springer-Verlag, 1967.

[5] A. Misiak, n-inner product spaces, Math. Nachr 140 (1989), 200–319.

[6] B. S. Reddy, Some Properties in Generalized n-inner product spaces,Int. Journal of Math. Analysis 4 (2010), 2229–2234.

[7] S. Roman, Advanced Linear Algebra, third ed., Springer, 2008.

[8] K. Trenˇcevsky and R. Malˇceski, On generalized n-inner product and the corresponding Cauchy-Schwarz inequality, J. Inequal. Pure and Appl. Math. 7 Art. 57 (2006).

Received: November, 2012