The Projection Methods in Countably Normed Spaces Nashat Faried* and Hany a El-Sharkawy

Faried and El-Sharkawy Journal of Inequalities and Applications (2015)2015:45 DOI 10.1186/s13660-014-0540-0

R E S E A R C H Open Access The projection methods in countably normed spaces Nashat Faried* and Hany A El-Sharkawy

*Correspondence: [email protected] Abstract Department of Mathematics, Faculty of Science, Ain Shams It is well known that a normed space is uniformly convex (smooth) if and only if its University, Abbassia, Cairo, 11566, dual space is uniformly smooth (convex). We extend the notions of uniform convexity Egypt (smoothness) from normed spaces to countably normed spaces ‘in which there is a countable number of compatible norms’. We get some fundamental links between Lindenstrauss duality formulas. A duality property between uniform convexity and uniform smoothness of countably normed spaces is also given. Moreover, based on the compatibility of those norms, it is interesting to show that ‘from any point in a real uniformly convex complete countably normed space, the nearest point to a nonempty convex closed subset of the space is the same for all norms’, which is helpful in further studies for ﬁxed points.

1 Introduction Deﬁnition . (Uniformly convex space [–]) A normed linear space E is called uniformly convex if for any ∈ (, ] there exists a δ = δ()>suchthatifx, y ∈ E with x =, ≥  ≤ y =and x – y then  (x + y) –δ.

Deﬁnition . (Modulus of convexity [–]) Let E be a normed linear space with dim E ≥

. The modulus of convexity of E is the function δE :(,]→ [, ] deﬁned by x + y δE():=inf – : x≤, y≤; x – y≥ . 

Deﬁnition . (Uniformly smooth space [–]) A normed linear space E is said to be uniformly smooth if whenever given >thereexistsδ >suchthatifx =andy≤δ then

x + y + x – y <+y.

Deﬁnition . (Modulus of smoothness [–]) Let E be a normed linear space with

dim E ≥ . The modulus of smoothness of E is the function ρE :[,∞) → [, ∞)deﬁnedby x + y + x – y ρ (τ):=sup –:x =;y = τ E  x + τy + x – τy = sup –:x ==y . 

© 2015 Faried and El-Sharkawy; licensee Springer. This is an Open Access article distributed under the terms of the Creative Com- mons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and repro- duction in any medium, provided the original work is properly credited. Faried and El-Sharkawy Journal of Inequalities and Applications (2015)2015:45 Page 2 of 9

Let K be a nonempty convex subset of a real normed linear space E.Forstrictcontrac- tion self-mappings of K into itself, with a fixed point in K, a well known iterative method ‘the celebrated Picard method’ has successfully been employed to approximate such fixed points. If, however, the domain of a mapping is a proper subset of E (and this is the case in several applications), and it maps K into E, this iteration method may not be well de- fined. In this situation, for Hilbert spaces and uniformly convex uniformly smooth Banach spaces, this problem has been overcome by the introduction of the metric projection in the recursion formulas (see, for example, [–]).

Deﬁnition . (Metric projection [, , , ]) Let E be a real uniformly convex and uniformly smooth Banach space, K be a nonempty proper subset of E. The operator

PK : E → K is called a metric projection operator if it assigns to each x ∈ E its nearest point x¯ ∈ K, which is the solution of the minimization problem

PK x = x¯; x¯: x – x¯ = inf x – ξ. ξ∈K

It is our purpose in this paper to extend the notion of uniform convexity and uniform smoothness of Banach spaces to countably normed spaces. Moreover, by extending some theorems to the case of uniformly convex uniformly smooth countably normed spaces, we prove the existence and uniqueness of nearest points in these spaces. Our theorems generalize some results of [, , ].

2Preliminaries

Deﬁnition . (Countably normed space [, ]) Two norms · and · in a linear

space E are said to be compatible if, whenever a sequence {xn} in E is Cauchy with respect to both norms and converges to a limit x ∈ E with respect to one of them, it also converges to the same limit x with respect to the other norm. A linear space E equipped with a

countable system of compatible norms ·n is said to be countably normed.Onecan prove that every countably normed linear space becomes a topological linear space when equipped with the topology generated by the neighborhood base consisting of all sets of the form Ur, = x : x ∈ E; x < ,...,xr <

for some number >  and positive integer r.

| | n Remark . ([]) By considering the new norms x n = maxi= x i we may assume that the sequence of norms { · n; n =,,...} is increasing, i.e.,

x ≤x ≤···≤xn ≤··· , ∀x ∈ E.

If E is a countably normed space, the completion of E in the norm ·n is denoted by En.

Then, by deﬁnition, En is a Banach space. Also in the light of Remark ., we can assume that

E ⊂···⊂En+ ⊂ En ⊂···⊂E. Faried and El-Sharkawy Journal of Inequalities and Applications (2015)2015:45 Page 3 of 9

Proposition . ([]) Let E be a countably normed space. Then E is complete if and only ∞ if E = n= En.

∗ Each Banach space En has a dual, which is a Banach space and denoted by En. ∗ ∞ ∗ Proposition . ([]) The dual of a countably normed space E is given by E = n= En andwehavethefollowinginclusions:

∗ ⊂···⊂ ∗ ⊂ ∗ ⊂···⊂ ∗ E En En+ E .

∈ ∗ ≥ Moreover, for f En we have f n f n+.

Remark . A countably normed space is metrizable and its metric d can be deﬁned by ∞ d(x, y)=  x–y i . i= i +x–yi

Example . An example of a countably normed space is the space of entire functions that are analytic in the unit disc |z| <  with the topology of the uniform convergence on

any closed subset of the disc and with the collection of norms x(z)n = max| |≤  |x(z)|. z – n Example . ∞ p+ q For  < p < ,thespace := q>p is a countably normed space. In fact, p+ pn one can easily see that = n for any choice of a monotonic decreasing sequence pn {pn} converging to p. Using Proposition . and the fact that is Banach for every n,it is clear now that the countably normed space p+ is complete.

3 Main results In this section, we give new deﬁnitions and prove our main theorems.

Deﬁnition . A countably normed space E is said to be uniformly convex if (Ei, ·i)

is uniformly convex for all i, i.e.,ifforeachi, ∀ >,∃δi()>suchthatifx, y ∈ Ei with ≥ x+y x i == y i and x – y i ,then–  i > δi.

If inf δi > , then one may call the space E equi-uniformly convex.

Deﬁnition . A countably normed space E is said to be uniformly smooth if (Ei, ·i)is

uniformly smooth for all i, i.e.,ifforeachi whenever given >thereexistsδi >such

that if xi =andyi ≤ δi then

x + yi + x – yi <+yi.

Proposition . A countably normed linear space E is uniformly convex if and only if for ∈ each i we have δEi ()>for all (, ].

Proof Assume that (Ei, ·i) is uniformly convex for all i.Then,foreachi,given > ≤ x+y there exists δi >suchthatδi –  i for every x and y in Ei such that x i == y i ≥ ≥ and x – y i . Therefore δEi () δi >foralli. ∈ ∈ Conversely, assume that for each i, δEi () >  for all (, ]. Fix (, ] and take ≥ ≤ x+y x, y in Ei with x i == y i and x – y i ,then<δEi () –  i and therefore Faried and El-Sharkawy Journal of Inequalities and Applications (2015)2015:45 Page 4 of 9

x+y ≤ ·  i –δi with δi = δEi ()whichdoesnotdependonx or y.Then(Ei, i)isuniformly convex for all i and hence the countably normed space E is uniformly convex.

In Proposition ., we showed the conditions of equivalence to uniform convexity of countably normed spaces, and now in Theorem ., we show the conditions of equivalence to uniform smoothness of countably normed spaces.

Theorem . A countably normed space E is uniformly smooth if and only if

ρ (t) lim Ei =, ∀i. t→+ t

Proof Assume that (Ei, ·i) is uniformly smooth for each i and if > , then there exists x+y i+ x–y i δi >suchthat  –<  y i for every x, y in Ei with x i =and y i = δi.This implies that for each i,wehaveρEi (t)<  t for every t < δi. ρ Ei (t) Conversely, for each i,given >  suppose that there exists δi >suchthat t <  for every t < δi.Letx, y be in Ei such that xi =andyi = δi.Thenwitht = yi we have

x + yi + x – yi <+yi.Then(Ei, ·i) is uniformly smooth for all i and hence the countably normed space E is uniformly smooth.

Now, we prove one of the fundamental links between the Lindenstrauss duality formulas.

Proposition . Let E be a countably normed space and for each n, let En be the completion · ∈ ∗ ∈ ∗ of E in the norm n. Then for each i we have: for every τ >,x Ei, x i =,and x Ei ∗ with x i =, τ ∗ ρE (τ)=sup – δ ():< ≤  . i  Ei

∗ ∗ ∈ ∗ ∗ ∗ Proof Let τ >andletx , y Ei with x i = y i = . For any η > , from the deﬁnition · ∗ ∈ of i in Ei there exist x, y Ei with x i = y i =suchthat ∗ ∗ ≤ ∗ ∗ ∗ ∗ ≤ ∗ ∗ x + y i – η x, x + y i, x – y i – η y, x – y i.

Using these two inequalities together with the fact that in Banach spaces we have xi = ∗ ∗ sup{| x, x i| : x i =},wehave ∗ ∗ ∗ ∗ x + y i + τ x – y i –

≤ ∗ ∗ ∗ ∗ x, x + y i + τ y, x – y i –+η( + τ)

∗ ∗ = x + τy, x i + x – τy, y i –+η( + τ)

≤x + τyi + x – τyi –+η( + τ) ≤ sup x + τyi + x – τyi –:xi = yi = + η( + τ)

=ρEi (τ)+η( + τ). Faried and El-Sharkawy Journal of Inequalities and Applications (2015)2015:45 Page 5 of 9

∗ ∗ If  < ≤x – y i ≤ , then we have ∗ ∗ τ η ≤ x + y – ρEi (τ)– ( + τ) – ,    i

which implies that

τ η ∗ – ρE (τ)– ( + τ) ≤ δ ().  i  Ei

Since η is arbitrary we conclude that

τ ∗ – ρE (τ) ≤ δ (), ∀ ∈ (, ]  i Ei τ ∗ ⇒ sup – δ (): ∈ (, ] ≤ ρE (τ).  Ei i

On the other hand, let x, y be in Ei with xi = yi =andletτ > . By the Hahn-Banach ∗ ∗ ∈ ∗ ∗ ∗ theorem there exist x, y Ei with x i = y i =andsuchthat

∗ ∗ x + τy, x i = x + τy i, x – τy, y i = x – τy i.

Then

∗ ∗ x + τy i + x – τy i – = x + τy, x i + x – τy, y i –

∗ ∗ ∗ ∗ = x, x + y i + τ y, x – y i – ≤ ∗ ∗ ∗ ∗ x + y i + τ y, x – y i –.

| ∗ ∗ | ≤ ≤ Hence, if we deﬁne  := y, x – y i ,then< x – y i and

x + τy + x – τy x∗ + y∗ + τ| y, x∗ – y∗ | i i –≤   i   i –   τ x∗ + y∗ =  – –   i   τ  ∗ ≤ – δ ()  Ei τ ≤ sup – δ ∗ ():< ≤  .  Ei

Therefore, τ ∗ ρE (τ) ≤ sup – δ ():< ≤  . i  Ei

The following two theorems give or determine some duality property concerning uniform convexity and uniform smoothness of countably normed spaces.

Theorem . Let E be a countably normed space, then

⇐⇒ ∗ Eisuniformlysmooth Ei is uniformly convex for all i. Faried and El-Sharkawy Journal of Inequalities and Applications (2015)2015:45 Page 6 of 9

Proof We will prove both directions by contradiction.

∗ ⇒ · ∗ ‘ ’ Assume that (Ei , i ) is not uniformly convex for some i. Therefore, δE ()=  i for some  ∈ (, ]. Using Proposition .,wegetforanyτ >,

ρE (τ) ρE (τ) <  ≤ i , hence lim i =,  τ τ τ

which shows that E is not uniformly smooth. ‘⇐’ Assume that E is not uniformly smooth, then

ρ Ei (t) ∃i: lim =, t→+ t

this means that there exists >such that for every δ >we can ﬁnd tδ with 

and ρ (tδ) ≥ tδ. Consequently, one can choose a sequence (τ ) such that <τ <, Ei n n τ → ,andρ (τ ) ≥ τ > τ . Using Proposition ., for every n there exists ∈ n Ei n n  n n (, ] such that

τ n n ∗ τn ≤ – δE (n),   i

which implies

τ ∗ n <δE (n) ≤ (n – ), i 

∗ in particular < n and δE (n) → . Recalling the fact that δE∗ is a nondecreasing i ∗ ∗ ∗ function we get δE () ≤ δE (n) → . Therefore E is not uniformly convex. i i

The proof of the following theorem is easy.

Theorem . Let E be a countably normed space, then

⇐⇒ ∗ Eisuniformlyconvex Ei is uniformly smooth for all i.

The following example is a direct application on the previous theorems.

Example . (Uniformly convex and uniformly smooth countably normed space) It is well  { ∈ R ∀ ∈ N ∞ | | ∞} known that the space = (xn):xn , n , n= xn < with the norm (xn)  = ∞ | | n= xn is uniformly convex normed space. Besides, it is uniformly smooth, because ()∗ = .    On , we deﬁne a countable number of seminorms by pi,i+((xn)) = xi + xi+,where  i =,,,.... Alsodeﬁninga countablenumberofcompatiblenormson by (xn)i =  pi,i+((xn)) + (xn).Then( , { · i, i =,,,...}) is a countably normed space.  Now, it is clear that ( , { · i, i = ,,,...}) is uniformly smooth (convex) countably   · normed space, as its completion i =( , i) is uniformly smooth (convex) complete normed space for all i. ρ (t)  ≤ → + · Moreover, ρ (t) ρ (t)+ρ (t). Then limt  t =because( , )isuni- i pi,i+ ρ (t)  pi,i+   → + R formly smooth normed space. Also limt  t =because( , pi,i+)lookslike Faried and El-Sharkawy Journal of Inequalities and Applications (2015)2015:45 Page 7 of 9

 with the Euclidean norm and it is uniformly smooth although ( , pi,i+)isseminormed space. Therefore,

ρ (t) lim i =, ∀i. t→+ t

Proposition . Let E be a countably normed space, {xn} be a sequence in E. Then {xn} is

aCauchysequencein(E, d) if and only if it is a Cauchy sequence in (E, ·i) for all i. We have

∞  x – x d(x , x )= n m i →  ⇐⇒ x – x → , ∀i. n m i +x – x n m i i= n m i

Proof The ﬁrst direction is trivial.

Conversely, assume that xn –xmi → , ∀i.Then,foreachi, ∀ >,∃ni: n, m > ni implies xn – xm i <  . ∞ ∀ >,∃i such that  < .Letn = maxi n ,then∀ >,∃n such that x –  i=i+ i   i= i  n ∀ ≤ ∀ ≥ xm i <  , i i, n, m n. Therefore,

i   x – x n, m ≥ n ⇒ d(x , x )< n m i + < + .  n m i +x – x    i= n m i

In the following theorem, we establish one of the most important and interesting geometric property in uniformly convex countably normed spaces, the metric projection point x¯ ‘the solution of the minimization problem’ is well deﬁned, that is to say, we have ‘existence and uniqueness’ for all norms.

Theorem . Let E be a real uniformly convex complete countably normed space, Kbe

a nonempty convex proper subset of E such that K is closed in each Ei. Then the metric projection is well deﬁned on K, i.e.,

∀x ∈ E \ K, ∃!x¯ ∈ K: x – x¯i = inf x – ξi, ∀i. ξ∈K

Claim Let E be a real uniformly convex Banach space. If {xn} is a sequence in E:

(a) limn→∞ xn =and (b) limn,m→∞ xn + xm =.Then {xn} is a Cauchy sequence in (E, ·).

Proof of the claim Suppose contrarily that the sequence {xn} is not a Cauchy sequence in (E, ·), then

∃: ∀N ∈ N, ∃m, n ∈ N: m, n > N while xn – xm≥.()

(E, ·) being uniformly convex implies that x + y ∀ >,∃δ >:∀x, y ∈ E: x≤, y≤, x – y≥ ⇒ ≤ –δ.()  Faried and El-Sharkawy Journal of Inequalities and Applications (2015)2015:45 Page 8 of 9

≤ xn →  →∞ Let  M <beﬁxed,then M M <asn .Then xn xm ∃n ∈ N: n, m ≥ n ⇒ ≤ ; ≤ , M M

xn–xm ≥ ∀ ≥ ∀ ≥ now ()implies M M >  , n, m n, i.e., N n.Hence,()implies xn + xm ≤ –δ ⇒xn + xm≤( – δ)M, ∀N ≥ n.() M

As N →∞ ⇒ n, m →∞, therefore xn + xm→togetherwith(), we get  ≤ ( – δ)M. Without loss of generality, we may assume that M → . Then  ≤ –δ,whichis impossible because δ > . This contradiction ﬁnishes the proof of the claim.

Proof of Theorem . Since x ∈/ K and K is closed for each i,wehavedi := infξ∈K x –

ξi >. i i i x–ξn That is, there exists a sequence ξ ∈ K: limn x – ξ i = di, which implies limn i =. n n di

ξ i i x – n ∈ ⇒ i → →∞ For each i, un := Ei un i asn .() di

i i ξn+ξm ∈ ∀ i i ∈ ∀ ∈ N Since K is convex, then  K, ξn, ξm K, n, m .Then i i i i i i ξn + ξm x – ξn x – ξm x – ξn x – ξm di ≤ x – = + ≤ +      i i i i i i i i x – ξn x – ξm x – ξn x – ξm ⇒  ≤ + ≤ + di di i di i di i ⇒ ≤ i i ≤ i i  un + um i un i + um i, i i → →∞ un + um i asn, m .()

{ i } From ()and(), using the claim, then, for each i, un is a Cauchy sequence in Ei, i.e., i i i i ξn–ξm i u – u i → , which gives i → . Therefore, {ξ } is a Cauchy sequence in K ⊂ n m di n ⊂ · i → i ∈ · →∞ E Ei.SinceEi is a complete space for each i,foreachi, ξn ξ Ei in i as n . i j Compatibility of norms implies that the limit ξ ∈ Ei is equal to the limit ξ ∈ Ej for all i = j, ξ i → ¯ ∈ · →∞ i.e.,foreach i, n x Ei in i as n .SinceE is complete, using Proposition ., ¯ ∈ i → ¯ ∈ E = i Ei, therefore there exists x E such that ξn x E.SinceK is closed in each · i ∈ ∀ ¯ ∈ i → ¯ · (Ei, i)andξn K, n,wehavex K,hencex – ξn x – x in each i.Usingthe · i → ¯ i → continuity of the i function, then x – ξn i x – x i.Weknowthat x – ξn i di, so from the uniqueness of the limit, we get x – x¯i = di: x¯ ∈ K. ∗ ∗ ∈ ∗ ∗ ¯ x¯+x ∈ Now we prove the uniqueness: Assume that x K: x–x i = di and x = x.Since  K, because of the convexity of K, ¯ ∗ ¯ ∗ x + x x – x x – x di di di ≤ x – ≤ + = + = di,  i  i  i  

∗ ∗ x¯+x x¯+x ∗ i.e., x– i = di where x¯ = = x . This contradicts the fact that in a uniformly convex   ∗ ∗ ∈ { ≤ } y+y space E, for all y = y B(x, δ):= x : x – x δ ,wemusthave x –  < δ. Faried and El-Sharkawy Journal of Inequalities and Applications (2015)2015:45 Page 9 of 9

Competing interests The authors declare that they have no competing interests.

Authors’ contributions The authors contributed equally to the manuscript and read and approved the ﬁnal manuscript.

Author details Department of Mathematics, Faculty of Science, Ain Shams University, Abbassia, Cairo, 11566, Egypt.

Acknowledgements The authors would like to thank the editor and the referees for their valuable comments.

Received: 9 April 2014 Accepted: 20 December 2014

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