p-summable SEQUENCE SPACES WITH INNER PRODUCTS

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1 p-summable SEQUENCE SPACES WITH INNER PRODUCTS HENDRA GUNAWAN, SUKRAN KONCA AND MOCHAMMAD IDRIS 3 Abstract. We revisit the space l p of p-summable sequences of real numbers. In particular, we show that this space is actually contained in a (weighted) inner product space. The relationship between l p and the (weighted) inner product space that contains l p is studied. For p >, we also obtain a result which describe how the weighted inner product space is associated to the weights.. Introduction By l p = l p (R) we denote the space of all p-summable sequences of real numbers. We now that for p, the space l p is not an inner product space, since the usual norm x p = ( = x p ) p on l p does not satisfy the paralellogram law. As an infinite dimensional normed space, l p can be equipped with another norm which is not equivalent to the usual norm. For example, x := ( = x p ) p, is such a norm (see [4]). However, one may observe that this norm does not satisfy the paralellogram law too for p. One question arises: can we define a norm on l p which satisfies the paralellogram law? The reason why we are interested in the paralellogram law is because we eventually wish to define an inner product, possibly with weights, on l p, so that we can define orthogonality and many other notions on this space. One alternative is to define a semi-inner product on l p as in [6], but having a semi-inner product is not as nice as having an inner product. The reader might agree that inner product spaces, which were initially introduced by D. Hilbert [3] in 9, have been up to now the most useful spaces in practical applications of functional analysis. In this paper, we introduce a (weighted) inner product on l p where p <. We discuss the properties of the induced norm and its relationship with the usual norm on l p. We also find that the inner product is actually defined on a larger space. We study the relationship between l p and this larger space, and found many interesting results, which we shall present in the following sections. Throughout the paper, we assume that X is a real vector space. As in [5], the norm on X is a mapping : X R such that for all vectors x, y X and scalars α R we have: () x 0 and x = 0 if and only if x = 0, Date: Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz. Corresponding author. 000 Mathematics Subect Classification. Primary 46C50; Secondary 46B0; 46C05; 46C5;46B99. Key words and phrases. Inner product spaces; normed spaces; p-summable sequences; weights.

2 HENDRA GUNAWAN, SUKRAN KONCA, MOCHAMMAD IDRIS () αx = α x, (3) x + y x + y. The inner product on X is a mapping, from X X into R such that for all vectors x, y, z X and scalars α we have: () x, x 0 and x, x = 0 if and only if x = 0; () x, y = y, x, (3) αx, y = α x, y, (4) x + x, y = x, y + x, y. As we wor with sequence spaces of real numbers, we will use the sum notation instead of, for brevity. =. Results for p In this section, we let p, unless otherwise stated. First, we observe that l p l (as sets). Indeed, if x = (x ) is a sequence in l p, then x = x = x p x p = sup x p ] p x p p x p ] p. Taing the square roots of both sides, we get ] x x p x p x p ] p, which means that x is in l. Thus we realize that l p can actually be considered as a subspace of l, equipped with the inner product x, y := x y, (.) and the norm ] x := x. (.) Being an induced norm from the inner product, the norm of course satisfies the paralellogram law: for every x, y l p. x + y + x y = x + y,

3 p-summable SEQUENCE SPACES WITH INNER PRODUCTS 3 A more general result is formulated in the following proposition, which describes the monotonicity property of the norms on l p spaces. Proposition.. If p q, then l p l q, with x q x p for every x l p. Moreover, the inclusion is strict: if p < q, then we can find a sequence x in l q which is not in l p. Proof. Let p q infty. For every x l p, we have x q q = x q = x q p x p = sup x q p ] q p x p p Taing the q-th roots of both sides, we get x p ] q p. x q x p, x p x p which tells us that l p l q. To show that the inclusion is strict, one may tae x = (x ) := ( ), where /p p <. Clearly x l q for p < q, but x / l p. When we equip l p, where p <, with, one might as whether is equivalent with the usual norm p. The answer is negative. We already have x x p for every x l p. The following proposition tells us that we cannot control x p by x for every x l p. Proposition.. Let p <. There is no constant C > 0 such that x C x p for every x l p. ( ) Proof. For each n N, tae x (n) :=. Then, p + n x (n) = p + n p < (independent of n), while x (n) p p = + p n < (dependent of n), which tends to as n. Hence x (n) 0, x (n) p as n.

4 4 HENDRA GUNAWAN, SUKRAN KONCA, MOCHAMMAD IDRIS So, there is no constant C > 0 such that x C x p for every x l p. Proposition.3. Let p <. As a set in (l, ), l p is not closed but dense in l. Proof. ( To show that l p is not closed in l, for each n N we tae x (n) :=,,...,, 0, 0,... ). Clearly (x (n) ) converges to x := ( ) /p n /p in. But /p x (n) l p for each n N, while x / l p. Therefore l p is not closed in l p. To show that l p is dense in l, we observe that every x = (x ) l can be approximated arbitrarily close by x (n) := (x,..., x n, 0, 0,... ) l p, n N. Remar.4. Propositions. and.3 warn us that when we use the l -inner product and its induced norm on l p for p <, we have to be careful especially when we deal with the topology. 3. Results for < p < Throughout this section, we let < p <, unless otherwise stated. As we have seen in Proposition., the space l p is now larger than l. Thus the usual inner product and norm on l are not defined for all sequences in l p. To define an inner product and a new norm on l p which satisfies the paralellogram law, we have to use weights. Let us choose v = (v ) l p where v > 0, N. Next we define the mapping, v which maps every pair of sequences x = (x ) and y = (y ) to x, y v := v p x y, (3.) and the mapping,v which maps every sequence x = (x ) to ] x,v := v p x. (3.) We observe that the mappings are well defined on l p. Indeed, for x = (x ) and y = (y ) in l p, it follows from Hölder s inequality that ] p [ ] [ ] v p x y v p p x p p y p p. (3.3) Thus the two mappings are defined on l p. proposition, whose proof is left to the reader. Moreover, we have the following Proposition 3.. The mappings in (3.) and (3.) define a weighted inner product and a weighted norm, respectively, on l p. From (3.3), we see that the following inequality x,v v p p x p holds for every x l p. It is then tempting to as whether the two norms are equivalent on l p. The answer, however, is negative, due to the following result.

5 p-summable SEQUENCE SPACES WITH INNER PRODUCTS 5 Proposition 3.. There is no constant C = C v > 0 such that for every x l p. x p C x,v, Proof. Suppose that such a constant exists. Then, for x := e n = (0,..., 0,, 0,... ), where the is the n th term, we have C v p n. But this cannot be true, since v n 0 as n. According to Proposition 3., it is possible for us to find a sequence in l p which is divergent with respect to the norm p, but convergent with respect to the norm,v. Example 3.3. Let x (n) := e n l p, where e n = (0,..., 0,, 0,... ) (the is the n th term), then x (m) x (n) p = p 0 as m, n. Since (x (n) ) is not a Cauchy sequence with respect to p, it is not convergent with respect to p. However, x (n),v = v p n 0 as n, since v n 0 as n. Hence, (x (n) ) is convergent with respect to the norm,v. If we wish, we can also define another weighted norm β,v on l p, where β p <, by ] x β,v := v p β x β β. Here p may be less than. Note that if β = p, then β,v = p. The following proposition gives a relationship between two such weighted norms on l p. p(γ β) γ Proposition 3.4. Let β < γ p. Then we have x β,v v p for every x l p. Proof. Suppose that x l p. We compute ] x β,v = v p β x β β = v p ] γ β γβ [ p(γ β) = γβ v p x γ,v, p(γ β) γ v ] v p γ x γ γ (p γ)β ] γ v x β β x γ,v as desired.

6 6 HENDRA GUNAWAN, SUKRAN KONCA, MOCHAMMAD IDRIS Corollary 3.5. If β < < γ p, then there are constants C,v, C,v > 0 such that C,v x β,v x,v C,v x γ,v for every x l p. 4. Further Results Let < p <. As we have seen in the previous section, every sequence x l p has x,v <. This suggests that l p lives inside a larger space, consisting all sequences x with x,v <. Let us denote this space by l v. We shall now discuss some properties of this space. First, we have the following proposition, which describes the relationship between l v and l p. Proposition 4.. As sets, we have l p l v and the inclusion is strict. Proof. Let x l p. It follows from (3.3) that x,v v p p x p, which means that x l v. To show that the inclusion is strict, we need to find x = (x ) such that vp x < but x p =. We now that v > 0 for all N, and v 0 as. So, choose m N such that vm p <, m > m such that vm p <, m 3 > m such that vm p 3 <, and so on. Since the process never 3 stops, we obtain an increasing sequence of nonnegative integers (m ) such that vm p < for every N. Now put x := for = m, m, m 3,... and x := 0 otherwise. Hence v p x = vm p < =, while x p = =. This means that x is in l v but not in l p. Theorem 4.. The space (l v,,v ) is complete. Accordingly, (l v,, v ) is a Hilbert space. Proof. It is easy to see that the space (l v,,v ) is a linear normed space, so we omit the details. To prove the completeness, let (x (n) ) be any Cauchy sequence in the space l v, where x (n) = (x (n) ) = (x(n), x (n),... ). Then for every ε > 0 there is an n 0 N such that for all n, m > n 0 ] x (n),v = v p < ε. (4.) It follows that for each N we have v p v p = x (n),v < ε. Then εv p

7 p-summable SEQUENCE SPACES WITH INNER PRODUCTS 7 for all m, n > n 0. Thus, for each fixed N, (x (n) ) is a Cauchy sequence of real numbers. Hence, it is convergent, say x (n) x as n. Using these infinitely many limits x, x, x 3,..., we define the sequence x := (x, x, x 3,... ). Then we have constructed a candidate limit for the sequence (x (n) ). However, so far, we only have that each individual component of x (n) converges to the corresponding component of x, i.e., (x (n) ) converges componentwise to x. To prove that (x (n) ) converges to x in norm, we go bac to (4.) and pass it to the limit m. We obtain ] x (n) x,v = v p x < ε, for all n > n 0. Since the space l v is linear, we also get x = (x x n ) + x n l v. This completes the proof. The following proposition tells us that l p is not too far from l v. Proposition 4.3. As a set in (l v,,v ), l p is not closed but dense in l v. Proof. As in the proof of Proposition 4., we construct an increasing sequence of nonnegative integers (m ) such that vm p < for every N. Next, for each N, we define x (n) = (x (n) ) by x(n) := for = m, m,..., m n and x (n) := 0 otherwise. Then we see that (x (n) ) forms a Cauchy sequence in l v since for m > n we have m x (n),v = v p vm p < 0, m >n >n =n+ as n. Since l v is complete, (x (n) ) is convergent and we now that the limit is the sequence x = (x ) where x = for = m, m, m 3,... and x = 0 otherwise. While x (n) l p for every n N, we see that the limit x / l p. This shows that l p is not closed in l v. The fact that l p is dense in l v is easy to see, since every x = (x ) l v can be approximated by x (n) := (x,..., x n, 0, 0,... ) for sufficiently large n N. Proposition 4.3 motivates us to study l v further as the ambient space, replacing l p. So far, we have fixed the weight v = (v ). We now would lie to now how the space l v depends on the choice of v. Let V p be the collection of all sequences v = (v ) l p with v > 0 for every N. Let v = (v ), w = (w ) V p. We say that v and w are equivalent and write v w if and only if there exists a constant C > 0 such that C v w C v for every N. Our final theorem is the following.

8 8 HENDRA GUNAWAN, SUKRAN KONCA, MOCHAMMAD IDRIS Theorem 4.4. Let v, w V p. Then, the following statements are equivalent: () v w. () There exists a constant C > 0 such that and x,w C x,v, x l v, x,v C x,w, x l w. (3) l v = l w as sets, and the two norms,v and,w are equivalent there. (4) There exists a constant C > 0 such that C x,v x,w C x,v, x l p. Proof. The chain of implications () () (3) (4) are clear. Hence it remains only to show that (4) (). Assume that there exists a constant C > 0 such that C x,v x,w C x,v, x l p. Tae x := e n, where n N is fixed but arbitrary. Then x l p, so that x is in l v as well as in l w. Moreover, x,v = v p n and x,w = w p Hence, from our assumption, we obtain C v p n w p n Cv p n, and this holds for every n N. Taing the ( ) p -th roots, we conclude that v w. This completes the proof. n. Remar 4.5. Theorem 4.4 is interesting in the sense that the two spaces (l v,,v ) and (l w,,w ) are identical if and only if,v and,w are equivalent on l p, and this can be checed through the equivalence of v and w. Thus, for example, the space associated to u := ( ) ( is identical with that associated to v := +), but is different from the one associated to w := ( ). We also note that, according to Proposition 4.3, even though l v and l w may be different, both spaces contain l p as a dense subset. 5. Concluding Remars We have shown that the space l p can be equipped with a (weighted) inner product and its induced norm. Using the inner product, one may define orthogonality on l p, carry out the Gram-Schmidt process to get an orthogonal set, define the volume of an n-dimensional parallelepiped on l p (see []), and so on. When we have to deal with topology, however, we are suggested to consider the (weighted) inner product space that contains l p, namely l for p or l v for < p < (where v l p with v > 0 for every N), which are complete.

9 p-summable SEQUENCE SPACES WITH INNER PRODUCTS 9 There we might also be interested in bounded linear functionals. For example, for < p <, the functional f y :=, y v is linear and bounded on l v, and its norm can be given by f y (x) x, y v f y = sup = sup. x,v 0 x,v x,v 0 x,v y y,v Clearly f y y,v, and by taing x := we obtain f y = y,v. Moreover, we can prove an analog of the Riesz-Fréchet Theorem (see []), which states that for any bounded linear functional g on l v, there exists a unique y l v such that g(x) = x, y v for every x l v and g = y,v. All the results for < p < also hold for L p (R), the space of p-integrable functions on R. For p <, however, different situations tae place. References [] Berberian, S.K. (96): Introduction to Hilbert Space. New Yor: Oxford Univ. Press. [] Gunawan, H.; W. Setya-Budhi; Mashadi; S. Gemawati (005): On volumes of n- dimensional parallelepipeds on l p spaces, Univ. Beograd. Publ. Eletrotehn. Fa. Ser. Mat. 6, [3] Hilbert, D. (9): Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. Repr. 953, New Yor: Chelsea. [4] Idris, M.; S. Eariani; H. Gunawan (03): On the space of p-summable sequences. Mat. Vesni. 65:, [5] Kreyszig, E. (978): Introductory Functional Analysis with Applications. New Yor: John Wiley & Sons. [6] Miličić, P.M. (987): Une généralisation naturelle du produit scaleaire dans un espace normé et son utilisation. Publ. Inst. Math. (Beograd) 4:56, Department of Mathematics, Institut Tenologi Bandung, Bandung 403, Indonesia. address: hgunawan@math.itb.ac.id Department of Mathematics, Saarya University, 5487, Saarya, Turey. address: sonca@saarya.edu.tr 3 Department of Mathematics, Institut Tenologi Bandung, Bandung 403, Indonesia. address: 3 mochidris@students.itb.ac.id