Microsoft Word - 337-343 IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1806 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics |337    Q uasi-inner product spaces of quasi-Sobolev spaces and their completeness Jawad Kadhim Khalaf Al-Delfi jawadaldelfi@uomustansiriyah.edu.iq Dept. of Mathematics / College of Science / Al-Mustansiriyah University Abstract Sequences spaces ℓ , m ∈ ℝ , p ∈ ℝ tha𝑡 have called quasi-Sobolev spaces were introduced by Jawad . K. Al-Delfi in 2013 [ 1[ . In this paper, we deal with notion of quasi- inner product space by using concept of quasi-normed space which is generalized to normed space and given a relationship between pre-Hilbert space and a quasi-inner product space with important results and examples. Completeness properties in quasi-inner product space gives us concept of quasi-Hilbert space. We show that , not all quasi- Sobolev spaces ℓ , are quasi-Hilbert spaces. The best examples which are quasi- Hilbert spaces and Hilbert spaces are ℓ , where m ∈ ℝ . Finally, propositions, theorems an examples are our own unless otherwise referred. Keywords: quasi-Sobolev space, quasi-Banach space, G𝑎teaux derivative , quasi-inner product space, quasi-Hilbert space. smooth quasi-Hilbert space. IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1806 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics |338    1. Introduction The family of sequence spaces ℓ , 1 < p < ∞ are normed space where, ℓ is the only inner product space in this family. Completeness of these spaces can be proved with respect to appropriate norms [2, 3]. Since the triangle inequality fails in the family of sequence spaces ℓ , 0 < p < 1 where, there is no norm for this range, then imply that it is not Banach space . For a sequence space ℓ , where 0 < p < 1 and others , many concepts were introduced . One of these concepts is a quasi- Banach space which is based on the definition of a quasi- norm [4]. A quasi- Banach space is a topological linear space [5]. In ]1[ , we were constructed a set of all sequence spaces of power real number m, m ∈ ℝ . The new spaces have called quasi-Sobolev spaces and have denoted by ℓ . We were proved that these spaces are quasi-Banach spaces in case 0 < p < ∞ and they are Banach spaces for 1< p < ∞ . In our work, we need study these spaces with other concepts such as a pre-Hilbert space and a quasi- inner product space (q. i .p) and their completeness. In normed spaces, mathematicians have used G𝑎teaux derivatives to introduce notion of quasi- inner product space and have investigated properties of this concept such as completeness, smoothness and others [6,7, 8] . This paper is devoted transference above ideology on quasi-normed space to given (q. i .p) and is studied the relationship between this notion and others, in order to study quasi-inner product spaces for ℓ and their completeness. The paper consists of two sections. Section one includes definitions of quasi- normed space and quasi-Banach space with some useful results which are needed in the section two. One of important theorems which is presented in this section is Jordan-van Neumann theorem. This theorem gives necessary and sufficient conditions to be generated by an inner product space. The second two presents a G𝑎teaux derivative that has big role to define many concepts, such as quasi- inner product space with completeness property of it. Also, this section shows that this functional is an inner product function in pre-Hilbert spaces. A space ℓ , for every m ∈ ℝ and p ∈ ℝ is a quasi-Hilbert space if it is a quasi-inner product space. Hence, with ℓ , we find spaces which are quasi-Hilbert spaces and are not Hilbert spaces , spaces neither quasi-Hilbert spaces nor Hilbert spaces and spaces are quasi-Hilbert spaces and Hilbert space. 2. Quasi-normed spaces of sequence spaces. This section contains notions such as quasi-normed space, a pre-Hilbert space and others with the relationship between them. Also, theorems and equations which are useful in section two are introduced. Definition 1.1. [4]: A quasi-norm ||.||q on vector space V over the field of real numbers ℝ is a function :||.||q V  0, ∞ with the properties: IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1806 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics |339    (1) 0||||,,0||||  vVvv qq ↔ v= 0. (2) |||||||||| vv qq   ,  ,Vv ℝ. (3)  |||||||||||| wvCwv qqq   v, w, ∊ V, where C 1 is a constant independent of v , w. A quasi-normed space is denoted by  ||.||, qV or simply V. A function ||.||q be a norm if C = 1, thus it is generalization of norm . Every norm function is quasi-norm. The converse does not hold, in general. Since every quasi-normed space V is a metric space by d(v, w) = |||| wvq  , then it is atopological linear space and the concepts of fundamental sequences and completeness in quasi-normed spaces are given [ 5]. A quasi- Banach space is a complete quasi-normed space. Definition 1.2. A symmetric linear functional on 2V is a functional L such that: (1) L(𝛽 v + 𝜇 w, u) = 𝛽 L(v ,u) +𝜇L(w, u) ; (2) L(v , w) = L(w , v),  𝛽, 𝜇 ∊ ℝ.,  𝑣, 𝑤, 𝑢 ∈ 𝑉. Remark 1.3. It is obvious, any inner product function satisfies definition 1.2 and generates a quasi - norm which is |||| vq = 𝑣, 𝑣 /  v ∊ V Lemma 1.4. In a pre-Hilbert space V , one has the equality:  4|||| wvq 4|||| wvq  = )||||||||(8 22 wv qq  v, w  v, w, ∊ V (1) Proof: Using remark 1.3, we get  2|||| wvq < v+ w, v + w > =  2|||| vq 2< v , w > + 2|||| wq ⇒ 2|||| wvq  = 22 |||||||| wv qq  + 4 v, w 22 |||||||| wv qq  + 4 𝑣, 𝑤 . Also,  2|||| wvq  2|||| vq 2 < v , w > + 2|||| wq ⇒ 4|||| wvq  = 22 |||||||| wv qq  4 v, w 22 |||||||| wv qq  + 4 𝑣, 𝑤 . Thus,  4|||| wvq 4|||| wvq  = )||||||||(8 22 wv qq  v, w and this is the desired result. Definition 1.5. [1]: Let λ ⊂ℝ is monotonically increasing sequence such that lim → λ = + ∞, quasi- Sobolev spaces are sequence spaces ℓ , where 0 < p < ∞ and m ∈ ℝ which are defined as : ℓ = { 𝑣 𝑣 ∶   1k 𝜆 |𝑣 | ∞ . IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1806 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics |340    When m = 0 then ℓ = ℓ , 0 < p < ∞ . Theorem 1.6. [1]: For every m ∈ ℝ and p ∈ ℝ a space ℓ , is a quasi-Banach space with the function : |||| vq   1k 𝜆𝑘 𝑚𝑝 2 |𝑣𝑘|p 1/𝑝 . We note that the constant C = p12 / for p ∊ (0, 1), and C = 1 for p ∊ [1, + ∞) . Theorem 1.7. (parallelogram equality) Let V be a pre-Hilbert space. Then ∀ v, w ∊ V, 2|||| wvq  + 2|||| wvq  = 2||||2 vq + 2||||2 wq (2) Proof: Since V be a pre-Hilbert space and 𝑣 , 𝑤 = 1 4 2|||| wvq  1 4 2|||| wvq  from remark 1.3 and proof of lemma 1.4 , then putting this function in equation (1) we obtain the desired result. Now , we introduce Jordan-van Neumann theorem in quasi- normed spaces. Theorem 1.8. ( Jordan – van Neumann ) A quasi-normed space V is a pre-Hilbert space iff equality (2) is satisfied by the quasi- norm of V. Proof: The proof of this theorem is very technical and proceeds in a way similar to its version in normed space (see [3]). The next example shows the importance of the parallelogram equality mentioned in the previous theorem. Example 1.9: Let v and w belong to the quasi-normed space ℓ / , where v = {vk}={0.1,0, 0, 0, …}, w = {wk} = {0, 0.2, 0, 0, …} and take {𝞴k} = {k}, 𝑘 ∈ℕ. Then we have: 2|||| wv  ½   1k 𝜆 |x 𝑦 | / = 0.4792627792275938 = 2|||| wv  ½ , so 2|||| wv  ½ + 2|||| wv  ½ = 0.9585255584551875, and, 2 2|||| v ½ + 2 2|||| w ½ = 0.482842712474619. It is clear that two sides of the equation (2 ) do not hold. Thus , ℓ / is not pre-Hilbert space. 3.Quasi-inner product spaces of sequence spaces A G𝑎teaux derivative is used to define many concepts, such as quasi- inner product function, and smooth quasi-Hilbert space with some important results and examples. Definition 2.1. IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1806 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics |341    Let V be a vector space over the field ℝ equipped with ||.||q . A G𝑎teaux derivative of |||| vq is a functional 𝛿 (v ,w) at v ∈ V in the direction w ∈ V which is defined as: 𝛿(v ,w) = ( δ 𝑣, 𝑤 + δ 𝑣, 𝑤 ) such that: 𝛿 𝑣, 𝑤 = lim → ℎ |||| hwvq  |||| vq , and δ 𝑥, 𝑦 = lim→ ℎ |||| hwvq  |||| vq , where h ∊ ℝ \ 0 . In similar way, we define 𝛿 (w ,v). G𝑎teaux derivatives 𝛿(v,w) and 𝛿(w,v) inspires the functionals 𝜏(v,w) = |||| vq 𝛿(v ,w) and 𝜏 (w , v) = |||| wq 𝛿 (w , v) sequentially . Definition 2.2 A G𝑎teaux derivative 𝜏 (v ,w) is said to be quasi-inner product function if 𝜏 (w , v) exists and the next equality is satisfied:  4|||| wvq 4|||| wvq  = 8 ( 2|||| vq 𝜏 (v ,w) + 2|||| wq 𝜏 (w,v) ),  v, w ∊ V (3) Similarly, 𝜏 (w , v) . A space V is said to be a quasi-inner product if both 𝜏 (v , w) and 𝜏 (w , v) are quasi-inner product functions . Lemma 2.3 For every positive integer p ≥ 1 and m ∈ ℝ, the functional 𝜏 (v ,w) in quasi-Sobolev spaces ℓ exists and is defined as : 𝜏 (v ,w) = pq v 2||||  k 𝜆𝑘 |𝑣𝑘| p 1 sng 𝑣𝑘 𝑤𝑘,  v ∊ ℓ s.t. |||| vq ∊ E , where, E = |||| vq : |||| vq 0 , 𝑃 1 |||| vq 0 , 𝑃 2 and sng 𝑣 = 1, 𝑣 0 0, 𝑣 0 1, 𝑣 0 . (4) Similarly, we define 𝜏 𝑤 , 𝑣 . Proof: In definition 2.1, we use properties of limits of functions and applying definition of a quasi-norm function of ℓ which is in theorem 1.6 with help of the binomial theorem, which is for every positive integer p, 𝑣 𝑤 𝑣 𝑤 , we get Eq. (4). Proposition 2.4. IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1806 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics |342    The existence of the limit in definition of G𝑎teaux functions is necessary condition , not sufficient, in order that any quasi-normed space be a quasi-inner product space. Proof Suppose V is a quasi-normed space. From definition 2.1, we observe that existence of δ (v ,w) and δ (v ,w) are connected by the limit on behavior of the quasi-norm as h → ±0. hence, 𝜏 (v ,w) is exist if this limit is exist. Also , with 𝜏 (w ,v) similarly. To explains above condition is not sufficiently, we take the example: Example 2.5: Suppose v ,w ∈ ℓ , where v = {vk} = {1,0, 0, 0, …}, w = {wk} = {1, 1, 0, 0, …} and take {𝜆k} = { √𝑘}, 𝑘 ∈ ℕ. Then, using lemma 2.3,we get 𝜏(v ,w)= 1, 𝜏 (w ,v)= 0.372884880824589 . Thus, 𝜏(v ,w) and 𝜏 (w ,v) are exist . However, equation (3) is not satisfied. Therefore, the space ℓ is not quasi-inner product space. Remark 2.6. If cases the values of p differ from those values considered in lemma 2.3, we have quasi- Sobolev spaces ℓ which are not quasi- inner product. For instance, in case p ∊ 0,1 , as it is shown in the example 1.9. Indeed, with the space ℓ / , 𝛿 (v ,w) and 𝛿 (w ,v) do not exist, since there is no limit as h → ±0 from definition 2.1. Then right hand in Eq. (3) is not finite, while left hand equal zero. Definition 2.7 A quasi-normed space V is smooth if 𝛿 𝑣, 𝑤 and 𝛿 𝑣, 𝑤 have one value. When V is smooth quasi-normed space , then 𝜏 (v,w) |||| vq lim→ ℎ |||| hwvq  |||| vq . Similarly, 𝜏 (w , v) . Proposition 2.8. Every pre-Hilbert space.is a quasi-inner product space. Proof: Let V is a pre-Hilbert space. According to lemma 1.4, an inner product function gives eq. (1) . Also, By remark 1.3 and definition 2.1 , we obtain 𝜏(v ,w) = < v, w> and 𝜏(w ,v) = . Hence, we have equation (3), and the definition 2.2 is hold. Thus, V is an quasi-inner product space. The converse of proposition does not hold, consider the following example: Example 2.9: Take example 2.5 with replace space ℓ by ℓ . Since Eq. (3) is satisfied with quasi- normed space ℓ , where the left and right hand of Eq. (3) are equal to 16, so it is quasi-inner product space. But the left and right hand of Eq. (2) are not equal, hence this space is not a pre-Hilbert space. Definition 2.10. A complete quasi- inner product space is called a quasi-Hilbert space. If a quasi-Hilbert space is smooth, then it is called a smooth quasi-Hilbert space. We recall that completeness property is coming from this property of quasi-normed space. Theorem 2.11. IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1806 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics |343    For every 𝑚 ∈ ℝ, ℓ is a smooth quasi-Hilbert space and Hilbert space. Proof: According to lemma 2.3, we get 𝜏 (v,w) = k 𝜆𝑘 |𝑣𝑘| 𝑠𝑛𝑔 𝑣𝑘 𝑤𝑘,and 𝜏(w,v) = k 𝜆 𝑚|𝑤 | 𝑠𝑛𝑔 𝑤 𝑣 which are linear by definition 1.2, with definition of 𝜏 (v,w) and 𝜏(w,v) as above, then they are symmetric, that is, 𝜏(v,w) = 𝜏(w,v), and 𝜏(v ,v) = 2|||| vq 0, with equality iff v = 0. Hence, ℓ is a pre-Hilbert space. By proposition 2.8, it is a quasi-inner product space, where 8 k 𝜆 2𝑚|𝑣 | sng v w + 8  k 𝜆 |𝑤 | 𝑠𝑛𝑔 𝑤 𝑣 is value to both sides of equation (3). If we apply quasi-norm function of ℓ in definition 2.1, we obtain 𝛿 𝑣, 𝑤 = 𝛿 𝑣, 𝑤 since the limit in 𝛿 𝑣, 𝑤 itself one 𝛿 𝑣, 𝑤 . Then ℓ is smooth. Now, since ℓ is a quasi-Banach space for every 𝑚 ∈ ℝ by theorem 1.6, then it is complete under |||| vq = 𝜏 𝑣, 𝑣 / , i.e. every fundamental sequence {vk} , 𝑘 ∈ ℕ is convergent in it. Therefore, Theorem is proved. Remark 2.12. Since a space ℓ , for every m ∈ ℝ and p ∈ ℝ is a quasi-Banach space, then ℓ is a quasi-Hilbert space if it is a quasi-inner product space. References [1] J.K. Al-Delfi., Quasi-Sobolev Spaces ℓ ., Bulletin of South Ural State University , Series of “Mathematics.Mechanics . Physics”, 5, 1. , 107–109. ( In Russian). 2013. [2] W. Rudin, Functional Analysis., McGraw-Hill, Inc., New York, 1991. [3] A. H. Siddiqi, Functional Analysis With Applications., Tata McGraw-Hill Publishing Company, Ltd, New Delhi, India, 1986. [4] N. Kalton, Quasi-Banach Spaces., Handbook of the Geometry of Banach Spaces, Vol. Edit. by. Johnson W.B and. Lindenstrauss. J – Amsterdam etc.: Elsevier, 1099–1130. 2003 [5] J. Bergh ; J. Löfström, Interpolation Spaces. An Introduction., Berlin–Heidelberg– New York, Springer-Verlag, 1976. [6] P.M. Milicic , On the g-orthogonal projection and the best approximation of vector in a quasi- inner product spaces., Scientiae Mathematicae Japonicae, 4, 3, 941-944. 2001. [7] R .A . Tapia., A characterization of inner product spaces., Proc. Amer. Math. Soc., 41, 569-574. 1973 [8] A. Sahovi ; F.Vajzovi and Peco . S., Continuity conditions for the Hilbert transform on quasi-Hilbert spaces., Sarajevo journal of mathematics, (2014), Vol.10, No. 22, pp- 111–120.