Available at: http://www.ictp.trieste.it/ pub_off IC/2000/82

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

ROUGH CONVERGENCE IN NORMED LINEAR SPACES

H. X. Phu1

Institute of Mathematics, P.O.Box 631 Bo Ho, 10000 Hanoi, Vietnamand

The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

Abstract

x* G X is said to be an r-limit point of a sequence (XJ) in some normedlinear space (X, || • ||) if limsup||xj — x*|| < r (r > 0). The set of all r-limitpoints of (XJ), denoted by LIMrXj, is bounded, closed and convex. This paperdeals with further properties of LIMrXi in case X = lRn, in particular, with therelation between this rough convergence and other convergence notions, and withthe dependence of LIMrXj on the roughness degree r. For instance, the set-valuedmapping r i->- LIMrXj is strictly increasing and continuous on (f,+oo), wheref := inf{r G IR+ : LIMrXj ^ 0}. For a so-called p-Cauchy sequence (XJ) satisfying

Ve: > 0 3i£ : i7j > i£ =̂ > \\xi — Xj\\ < p + e,

it is shown that in general r = ^h[P (or r = . 2(n+i)P ^or Euclidean space) is the

best convergence degree such that LIMrXi ^ 0.

MIRAMARE - TRIESTE

July 2000

1 Regular Associate of the Abdus Salam ICTP.

1

1. INTRODUCTION

Let (xi) be a sequence in some normed linear space (X, || • ||), and r be a non-negative real number, (x^ is said to be r-convergent to x*, denoted by XjAx*,if

Ve: > 0 3i£ G IN : i > ie => \\xi - x*\\ < r + e, (1.1)

or equivalently, iflimsup \\xi — x*|| < r. (1-2)

This is the rough convergence with r as roughness degree. For r = 0 we havethe classical convergence again. But our proper interest is case r > 0. Thereare several reasons for this interest. For instance, since an originally convergentsequence (yi) (with yi —>• x*) often cannot be determined (i.e., measured or cal-culated) exactly, one has to do with an approximated sequence (xi) satisfying\\xi ~Vi\\ < r f°r a u h where r > 0 is the maximal approximation error. Then (xi)is no more convergent in the classical sense, but

\ \ x i - £ * | | < \ \ x i - y i \ \ + \\yi - £ * | | < r + \\yi - x * \ \

implies that it is r-convergent in the sense of (1-1).

If (1.1) holds, x* is an r-limit point of (x^), which is usually no more unique(for r > 0). So we have to consider the so-called r-limit set (or shortly: r-limit) of(xi) defined by

lAWxi := {x* G X : x^x*}. (1.3)

(x^ is said to be r-convergent if LIMrXj ^ 0. In this case, r is also called aconvergence degree of (xi).

Let us illustrate by an example. Clearly, the sequence (yi) with

2/, = 0.5 + 2 ( - l ) 7 i , « = 1,2,... (1.4)

converges to 0.5. For sufficiently large i, it is impossible to calculate yi exactly bycomputer, but it is rounded to some machine number, e.g. to the nearest one. Forthe sake of simplicity, we normalize so that yi is approximated by

Xi := rd(yi) := z where z is the integer satisfying z — 0.5 < yi < z + 0.5. (1.5)

Thenx\ = —1, X2 = 2, X2j-i = 0 and X2j = 1 for j = 2, 3 , . . .

Obviously, the sequence (xi) does not converge anymore. But by definition we

have XjA0.5 for r = 0.5, and altogether

r t v for r < 0.5L11V1 Xi — ^ r i 1 r ^ r, r

[1 — r, r\ for r > 0.5.

Sometimes we are interested in the set of r-limit points lying in a given subsetS C X, which is called r-limit in S and denoted by

LIMS'rXi := {x*eS: x^x*}. (1.6)

(Obviously, LIMx 'rXj = LIJVTXJ and LIMs'rXi = Sn LIMrXj.) For instance, withS = (XJ), the set of the r-limit points of (x^) lying in this sequence is considered,which is rather unusual in the classical point of view. In the example given by(1.4)-(1.5), we have

0 for r < 1{xH : i > 3} for r > 1.

Similarly as above, if some Cauchy sequence (yi) cannot be determined exactlybut is only approximated by a sequence (XJ) with p/2 > 0 as maximal approxi-mation error, i.e., \\x{ — yi\\ < p/2 for all i, then (xi) does not satisfy the classicalCauchy condition, but only fulfills the following rough Cauchy condition

vs > U dz£ : z, j > ie =>• ||Xj — Xj|| < p + e. (1.7J

Such a sequence is said to be a rough Cauchy sequence with roughness degree p, orp-Cauchy sequence for short, p is also called a Cauchy degree of (XJ).

Section 2 is devoted to some basic properties such as boundedness, closedness,convexity... of r-limit sets. In Section 3 the relation between rough convergenceand classical convergence or set convergence is investigated. The dependence of r-limit on the roughness degree r is the subject of Section 3. Section 4 is concernedwith rough Cauchy sequences, in particular, with the relation between Cauchydegree and convergence degree of a sequence.

For the sake of simplicity, assume throughout this paper that X = TRn.

2. ELEMENTARY PROPERTIES

First, let us transform some properties of classical convergence to rough con-vergence. It is well known if a sequence converges then its limit is unique. Thisproperty is not maintained for rough convergence with roughness degree r > 0,but only has the following analogy.

PROPOSITION 2.1. The diameter of a r-limit set is not greater than 2r. Ingeneral, there is no smaller bound.

Proof. We have to show that

dmm(LIMrXi) = sup{\\y - z\\:y,ze L I M r x J < 2r. (2.1)

3

Assume the contrary that diam(LIM r;Ej) > 2r, then there exist y,z E LIMrXisatisfying d := \\y — z\\ > 2r. For an arbitrary e E (0, d/2 — r ) , it follows from (1.1)and (1.3) that there is an ie E IN such that

\\xi ~ y\\ < r + £ a n ( i \\xi — z\\ < r + £ f° r ^ > i£.

This implies

Ib - z\\ < \\xi - y\\ + \\xi - z\\ < 2(r + s) < 2r + 2(d/2 - r) = d,

which conflicts with d = \\y — z\\. Hence, (2.1) must be true.

Consider a convergent sequence (xi) with l imxj = x*. Then, for

Br{x*) :={yEX: \\y - x*\\ < r} ,

it follows from

(1.1), and (1.3) that LIMr£j = Br(x*). Since diami?r(a:*) = 2r, this shows thatin general the upper bound 2r of the diameter of an r-limit set cannot be decreasedanymore. •

Obviously, the uniqueness of limit (of classical convergence) can be regarded asa special case of the latter property, because if r = 0 then diam( LIMra?j) = 2r = 0,i.e., LIMrXj is either empty or a singleton.

A further property of classical concept is the boundedness of convergent se-quences. Its analogy for rough convergence is:

PROPOSITION 2.2. A sequence (x^) is bounded if and only if there existsan r > 0 such that LIMrXj ^ 0. For all r > 0, a bounded sequence (x^) alwayscontains a subsequence (x^.) with

Proof. If s := sup{||xi|| : % E IN} < oo then LIMs£j contains the origin of X.On the other hand, if LIMr£j ^ 0 for some r > 0 then all but finite elements Xiare contained in some ball with any radius greater than r. Therefore, the sequence(xi) is bounded.

As (x^ is a bounded sequence in a finite-dimensional normed space, it cer-tainly contains a convergent subsequence (xij). Let x* be its limit point, then

ij = Br(x*) and, for r > 0,

^ ' r x Z j = {Xij : | | x * - Xij \ \ < r } ^ 0 .

•Note that the second part of the previous proposition is concerned with r-limit

points lying in the subsequence (x^). It is straightforward that a sequence con-tained in some bounded set S always possesses a subsequence being r-convergent(for an arbitrary r > 0) to some point of S. Here, the closedness of S is not neededas for classical convergence.

Corresponding to the property that each subsequence of a convergent sequencealso converges to the same limit point, we now have the following one whose proofis rather simple.

PROPOSITION 2.3. If (%[) is a subsequence of(xi), then LIMrXi C LIMrx-.

Those are possibly all properties having an analogy in classical concept. It isreally unimaginable to say something more about properties of classical limit setswhich are either empty or contain only a single point. But for rough convergence,the r-limit set is generally no more a singleton. So its topological and geometricalproperties are of interest, as given in the next propositions.

PROPOSITION 2.4. For all r > 0, the r-limit set LIMrXi of an arbitrarysequence (xt) is closed.

Proof. Let (jjj) be an arbitrary sequence in LIMr£j which converges to somepoint ?/*. For each e > 0, by definition, there are a j £ / 2 and an i£/2 such thatWVje/2 ~ V*\\ < e / 2 a n d \\xi — Vje/2\\ < r + e/2 whenever i > i£/2. Consequent ly ,

\\xi ~y*\\ < \\xi -Vje/iW + WVje/2 -V*\\<r + e if i> i£/2.

That means y* G LIMrXj, too. Hence, LIMrXj is closed. •

PROPOSITION 2.5

(a) 7/2/0 e LIMr°Xi and yx E L I M r i ^ then

yx := (1 - X)y0 + XVl E L I M ^ - ^ + ^ x , , for A E [0,1]. (2.2)

(b) LIMrXi is convex. If (X, || • ||) is a finite-dimensional strictly convex space(i.e., the closed unit ball is strictly convex) then LIMrXj is strictly convex,i.e., 2/o,I/i E LIMra?j and yo ^ y\ imply

yx E int(LIMrXi) for all A E (0,1).

Proof, (a) By definition, for every e > 0 there exists an ie such that i > ie

implies \\xi — yo\\ < ro + e a n d \\xi — yi\\ < r\ + e7 which yields also

\\xi - y\\\ < (1 - X)\\xi - j/o11 + X\\xi - yi\\

< ( l -

= ( 1 -

Hence, yx G L I M ( 1 - A ) r ° + A r i ^ .

(b) In particular, for r = TQ = r i , (a) yields immediately that LIMra?j is

convex.Assume the space considered is strictly convex. In order to prove the strict

convexity of LIMrXj it suffices to show that 2/0,2/1 G LIMra?j and 2/0 ^ 2/1 imply

2/0.5 = 2(2/0 + 2/1) e int(LIMrXj),

because for each 2/A5 0 < A < 1, there exist 2/0,2/i G LIMrXj satisfying y'o ^ 2/1 and

ULet C be the set of cluster points of (xi) which is obviously closed. Moreover,

since the normed space considered is finite-dimensional and (xi) is bounded (byProposition 2.2), C is nonempty and bounded. Therefore, there exists a c G Csatisfying

11 c — 2/0.511 = max ||c-2/O.5H-c£C

It follows from 2/0,2/1 G LIMrXi that

| | c — yo\\ < r a n d \\c — yi\\ < r

(compare (3.2)). These inequalities imply by the strict convexity of the spaceconsidered

||c-2/0.51| = ||0.5(c — 2/0) + 0 .5(c- | / i ) | | < max{||c-j/o| | , | | c - y i | | } < r,

and thus a := r — \\c — 2/0.511 > 0. Now we have for all c G C and y G Ba(y05)

\\c - y\\ < \\c - 2/0.51| + H2/0.5 - 2/|| < ||c - 2/0.5II + cr = r,

which implies by definition y G LIMr£j. That means 2/0.5 is an interior point ofLlMr

Xi.•

3. RELATION TO OTHER CONVERGENCE NOTIONS

In this section we investigate the relation between rough convergence and someother convergence notions, such as classical convergence and set convergence.

First, let us formulate an additive property of rough convergence.

PROPOSITION 3.1. Suppose r\ > 0 and r2 > 0. A sequence (xi) in X is(ri+r'2)-convergent to x* if and only if there exists a sequence (y^) in X such that

and \\xi — yi\\ < r2, i = 1, 2,... (3.1)

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Proof. Assume (3.1). y^x* means that for all e > 0 there exists an i£ suchthat

\\yi - x * | | < r\ + e if i> ie.

Since ||XJ — yi\\ < r2l this yields immediately

\\xi - x * | | < \\xi - y i \ \ + \\yi - x * | | < r i + r2 + e if i > i e -

Hence, (x^) is (ri + ^-convergent to x*.Assume now that (x^) is (ri + ^-convergent to x*. With

x* if Iki - ^*ll < r2j

we have

\\v.-r I K 1 ° if \\xi~x*W ^ r 2

//? *^ * l 11 II ' I ' l l 1 1 ^

I \\xi ~ X*\\ ~ r 2 !I | F i — a;*|| > ?*2and

Ik i - z / i l l < r 2 for * = i , 2 , . . .By (1.2), x , G UMri+r2Xi implies

limsup ||xi — x*|| < ri + r2,

and thereforelimsup \\yi - x * | | < r i ,

that means ^—Vx*. D

In particular, for n = 0 and r<i = r > 0, the latter conclusion says that asequence (XJ) is r-convergent to x* iff there exists a sequence (?/j) such that

Vi^-x* and ||XJ - yi\\ < r, z = 1, 2,...

The necessity means that if (XJ) is an approximation of a convergent sequenceHi —>• x* with r as maximal approximation error then it is still r-convergent to x*.This is what was already said in the introduction. On the other hand, if (XJ) isr-convergent to x* then there exists a sequence (yi) near (x^) (i.e., ||XJ — y^] < rfor all i) which converges (in the classical sense) to x*.

To state an example of use of the preceding, let us consider now a special typeof sequences which arise as integer part of others. It corresponds to the compu-tational practice where a real number is approximated by the maximal machinenumber less or equal to the exact one.

PROPOSITION 3.2. Assume (x^) is a sequence in (H n , || • ||) which convergesto x*. For x = ( x ^ x 2 , . . . ,xn) G IRn, denote [x] := ([x1], [x 2 ] , . . . , [xn]), where [a]is the integer part of the real number a.

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(a) / / || • || is the maximum norm then

x* G L I M 1 ^ ] and LIM°-5[x;] ^ 0.

(b) / / || • || is the Euclidean norm then

x* G LlM^[xi] and LIM°-5^[x;] ^ 0.

Proof. Since 0 < x\ — [xj] < 1 for alH G IN and j G {1 ,2 , . . . , n}, we have

1 if || • || is the maximal normlXi |X?:I" < ^ y/n if || • || is the Euclidean norm.

Therefore, Proposition 3.1 implies that

T T, ,r „ i 1 if || • || is the maximal norm* G L1M Xi lor r = { r- .c . , „ r j

n if • is the Euclidean norm.Let x* := [x*] — (0.5, 0 .5 , . . . , 0.5). Since Xj —> x*, there exists an i* such that

which yields [xj] G {[xi] — 1, [xi]}, and therefore

i r 31 ~ 7 r\ K f * /-~ r i o i * ^>

Hence0.5 if || • || is the maximal norm0.5\/n if || • || is the Euclidean norm

for i > i*. That means by definition

T T , , r r n c f 0.5 if || • || iG LIM \Xi\ lor r = < n _ ,- .c .

[ 0.5^/n it || • || i0.5 if || • || is the maximal norm

is the Euclidean norm.

•Note that all parameters r stated in Proposition 3.2 are already optimal, i.e.,

one cannot give smaller parameters which are suitable for any case. To see this,just consider the following.

EXAMPLE 3.1. Let

Then Xi = (x], £ ? , . . . , x") converges to x* = (0, 0 , . . . , 0), and

r -, _ J (0, 0 , . . . , 0) if i is even— (1, ! , . . . , ! ) if i is odd.

Since

1 if II • II is the maximal norm| | - ( l , l , . . . , l ) - ( O , O , . . . , O ) | | = . r .. ..

[ V n ^ II • II 1S the Euclidean norm,

it is easy to see that

x* £ UMr[Xi] for r< I1 if || • || is the maximal norm

ri if || • || is the Euclidean norm,

and

0.5 if || • || is the maximal normi] = ® for r < < n .

^ 0.5vn !i II " II 1S the Euclidean norm.

In the proof of Proposition 2.1 we have already shown that if (x^) convergesto x* then LIMrXj = Br(x*). This is a part of the next.

PROPOSITION 3.3. A sequence (x^ c H n converges to x* if and only ifUMr

Xi = Br(x*).

Proof. It remains to show that LIMra:j = Br(x*) implies Xi —> x*. Assumethe contrary that (x^ has a cluster point x'^ different from x*. Then the point

x* := xI F * — x;n

satisfies

Since x* is a cluster point, this inequality implies by definition that x* ^ LIMrXj,a contradiction to ||x*— x*|| = r and LIMrXi = Br{x*). Thus x* is the only clusterpoint of (XJ) as a bounded sequence (by Proposition 2.2) in some finite-dimensionalnormed space. Consequently, (x^) converges to x*. •

It follows directly that Xj —>• x* yields the existence of two points 2/1,2/2 £LIMrXj satisfying \\yi — y2\\ = 2r. In general, this existence does not imply theconvergence of (XJ), as the following shows.

EXAMPLE 3.2. Let (xi) be a sequence in IR2 defined by Xj := (&,0), ^ =(—1)*. For || • || as the maximum norm, we have

Clearly, 2/1 := (0,1) and 2/2 := (0, —1) satisfy 2/1,2/2 £ LIM1Xj and H2/1 — 2/2II = 2,but this sequence (xi) does not converge at all.

But this fact changes completely in strictly convex spaces.

9

PROPOSITION 3.4. Suppose (x^ is a sequence in some (finite-dimensional)strictly convex space. If there are 2/1,2/2 G LIMr£j satisfying \\yi — 2/2II = 2r then(xi) converges to §(2/1 + 2/2)-

Proof Let 2/3 be an arbitrary cluster point of (x^. Then 2/1,2/2 £implies

H2/1 -2 /3 | | < r and H2/2 — 2/31| < r

(compare Proposition 3.5). On the other hand,

2r = H2/1 - 2/2II < \\yi ~ 2/all + H2/2 - 2/3||-

Therefore

= 112/2 - 2 / 3 | | =r.

Since

1 1x(2/2 - 2/1) = ~ ((2/3 - 2/1) + (2/2 - 2/3)) a n d 0(2/2 - 2 / 1 )

the strict convexity of the normed space considered implies

2(2/2 - 2/i) = 2 / 3 - 2 / 1 = 2 / 2 - 2/3-

Hence, 2/3 = ^(2/1 + 2/2)- That means \{yi + 2/2) is the only cluster point of (xi) asa bounded sequence (by Proposition 2.2) in some finite-dimensional normed space.Consequently, (x^) must converge to \{yi +2/2)- D

The previous two assertions are concerned with the relation between a conver-gent sequence and its r-limit set. In general, we do not expect sequences consideredto be convergent, but to have several cluster points.

PROPOSITION 3.5.

(a) If c is a cluster point of the sequence (x^) then

UMrXi C Br(c). (3.2)

(b) Let C be the set of cluster points of (x^ C H n . Then

UMrXl = p | Br(c) = {x* G I T : C C Br{x*)}. (3.3)

c£C

Proof (a) For an arbitrary cluster point c of (xi) we have

||a;* - c|| < r for all x* G LIMrXj, (3.4)

otherwise there are infinite Xi satisfying

||x* — Xi\\ > r + s with e := (||x* — c\\ — r ) /2 > 0,

10

because c is a cluster point of (o?j), which contradicts (1.1). Hence, (3.2) must betrue.

(b) The preceding yields immediately

UMrXi C p| Br(c).

c£C

If y G ncecBr(c) then \\y — c\\ < r for all c G C, which is equivalent toCCBr(y),i.e.

f Br(c) C {x* G JRn : C C £ r (x*)}.

If y (£ LlMrXi then, by definition, there is an e > 0 such that there existinfinite Xi satisfying ||XJ — y\\ > r + e, which implies the existence of a cluster pointc of (xi) with \\y - c\\ > r + e, i.e., C % Br{y) and y g {x* G IRn : C C Br(x*)}.Hence, y G LIMrXj follows from y E {x* e IRn : C C B r(j; ,)}, i.e.

{x* G Mn : C C 5r(a;*)} C LIM r ^ .

The three inclusions proved above ensure that (3.3) holds true. •

As an example, consider again the sequence (xi) with Xi = ((—1)*,0) G IR2.Since it only has two cluster points (—1,0) and (1,0), it follows from (3.3) thatUMr

Xi = 5 r ( ( - l , 0)) n i? r((l , 0)).

(3.3) shows that an r-limit set of (xi) is "almost round" in such a sense that itsboundary consists of \C\ boundary parts of balls with radius r, where \C\ denotesthe cardinality of the cluster point set C of (x^).

Let us consider the relation of rough convergence to set convergence in Set-Valued Analysis. Recall if (Ki)^^ is a sequence of subsets of a metric space Xthen the subsets

fj := {x G X\ liminf d(x, K^) = 0},I—>-OO

Liminf Kt := {x E X\ lim d{x,Ki) = 0}i>oo itoo

are called upper or tower limit of the sequence (ifj) (see [1, p.17]).

By definition, Limsup{xj} is the set of cluster points of (XJ). Therefore, itfollows from (3.4)

Limsup{xj} C Br(x*) for all x* G LIMrXj,

and from (3.3)

Br(c),

cGLimsup {x{}

11

if the normed space considered is finite-dimensional.

PROPOSITION 3.6.

LIMrXi = Liminf Br(xi).

Proof, a) Assume y E LIMrXj. Define then

\ y otherwise.

Since

~xi) -y - 1 = \\\y- xi\\ ~\\y~xi\

we have

10 otherwise.

Therefore, y E LIMrXi yields that yi tends to y as i —> oo. But \\xi — yi\\ < r, i.e.,yi E Br(xi). Consequently, linij_j.0O d (?/, Br(xi)) = 0, which means by definitionthat y E Liminf Br(xi).

b) If y E Liminf 5 ( X J , r) then, by definition, there exists a sequence (yi)satisfying yi —>• y and yi E Br(xi)1 i.e., \\xi — yi\\ < r. Therefore, Proposition 3.1implies y E LIMr£j. •

4. DEPENDENCE ON ROUGHNESS DEGREE

The previous section is concerned with properties of r-limit set for a fixedroughness degree r. Let us now investigate the dependence of r-limit LIMro?j of afixed sequence (x^ on varying parameter r.

It follows from definition

LIMriXi C LIMr2Xi if n < r2. (4.1)

This monotonicity is included in the following.

PROPOSITION 4.1. Suppose r > 0 and a > 0. Then

(a) LIMrXi +Ba(0) <Z LIMr+<TXi.

(b) Ba{y) C LLVT^ im^es y E UMr~aXi.

Proof, (a) Let y E \AMrXi and z E Ba(0). By definition, for all e > 0 thereexists an z£ such that \\x{ — y\\ < r + e if i > iei which implies by \\z\\ < a that

\\xi — y — z\\<r + a + e if i > i£.

12

Hence, y + z G LlMr+aXi.

(b) Let c be an arbitrary cluster point of (XJ). If \\y — c\\ > r — a then thepoint

satisfies||x* — c\\ = a + \\y — c\\ > a + (r — a) = r.

By (3.2), this yields x* ^ LIM rXj, a contradiction to ||x* — y\\ = a and Ba{y) CLIMra;i. Hence, ||y — c|| < r — cr for all cluster points c G C. Consequently, itfollows from (3.3)

ye f |B r . f f (c)= \AMr-aXl.

cec

nIn general, LIMr£j + -BCT(0) ^ LIMr+<JXi is possible. For instance, the se-

quence (xi) in the two-dimensional Euclidean space with

^ = (0 ,&)elR 2 , & = (-!)% * = 1,2,...

satisfies

LIM0-5^ + 5o.5(O) = 0 + 50.5(0) = 0 + {(0, 0)} = LIM 1^.

The reason does not consist in the voidness of LIM0 5£j, because

LIM 1 ^ + 5i(0) = {(0, 0)} + Si(0) = Si(0)

while | |(^3,0) - xt\\ = \\(VH, ±1)|| = 2 for all i implies

G LIM2a;i \5i(0),

i.e., L I M ^ + S^O) ^ LIM 2^.

Definef := inf{r G 1R+ : L I M r ^ ^ 0}. (4.2)

By the monotonicity given in (4.1), we have

r f = 0 for r < f , ,LIM xi { . rt „ _ (4.3)

j / 0 f o r r > r . v y

Moreover, by Proposition 4.1, for all r > f and a G (0,r — f), LIMrXi alwayscontains some ball with radius cr, that means at least

in t (LIM r Xi)^0 for r > f. (4.4)

13

Therefore,

i n t ( L I M r ^ ) = 0 implies r<f and LlMr'xH = 0 for r'e[0,r). (4.5)

It remains to ask what propert ies the f-limit has.

P R O P O S I T I O N 4 .2 .

(a) r = f if and only if

UMrXi ^ 0 and int(LIMr£;) = 0. (4.6)

(b) / / (X, || • ||) is a finite-dimensional strictly convex space then r = f if and onlyif LIMrXi is a singleton.

Proof (a) Let r = f. We have to show (4.6). It follows from Proposition— / /

4.3 proved later that LIMrXi = f]r,>f LIMr Xi. For r' > f, LIMr Xi is nonempty

(by (4.3)) and closed (Proposition 2.4). By (4.1), we have

T Tlx/T' fy.. — [I T Tlx/T T •

i j l lVl X-i I i j l lVl Xq.% I I *'

r'>f f<r'<f+l

and LIMr Xi, r' G (f, f + 1] is a family of nonempty closed subsets in the compactset LIM r + Xi having the finite intersection property. Therefore, their intersectionis nonempty (see [2, p. 69]). Hence LIMr£j ^ 0.

If int( LIMrXi) ^ 0 then it contains some ball Ba(y) with a > 0, and by Propo-sition 4.1 we have LIMr~aXi ^ 0, i.e., r > f. Thus r = f yields int(LIMr;Ei) = 0.

Assume (4.6). Since LIMrXj ^ 0, we have r > f. On the other hand, by(4.5), r <f follows from int(LIMra?j) = 0. Consequently, r = f.

(b) If LIMrXi is a singleton then (4.6) is fulfilled. Therefore, by (a), r = f.It remains to show that LIMro?j is a singleton. This follows directly from its strictconvexity (by Proposition 2.5), LIMrXj ^ 0, and int(LIMrXj) = 0. •

Consider again the sequence (x^) with Xi = ((—l)l,0) G IR2. (3.3) yieldsLIM 1 ^ = .Bi((—1,0)) l~l J?i((l,0)). For both maximal and Euclidean norm, wehave

(0,0) G LIM 1 ^ ̂ 0 and int^LIM1^) = 0,

which implies by Proposition 4.2 that the minimal convergence degree of thissequence is f = 1. In particular, if || • || is the maximal norm then

This shows that the strict convexity of the space considered is really necessary forLIMrXi to be a singleton.

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Conversely, the uniqueness of r-limit point is no signal for the strict convexityof the space. For instance, consider (xi) C IR2 with

(0,1) if i = 4j + 1_ (1,0) if i = 4j + 2

1 S (0,1) if i = 4j + 3(0,1) if « = 4(i + 1)

(j = 0,1,...). Then for maximal norm, we also have LIM1^ = {(0, 0)}, althoughwith this norm the space is not strictly convex.

PROPOSITION 4.3.

cl ( ( J LlM^Xi) C LlMrXi = p | LIMr'xi.

0<r'<r r'>r

Ifr^f then Uo<r'<r LIMr'x, = UMrXl.

Proof. By the monotonicity given in (4.1) and the closedness of r-limit(Proposition 2.4) we have

cl ( [ J UMr>Xi) C UMrXi C p | LlMr'Xi.0<r'<r r'>r

Consider now an arbitrary y G X \ LIMr£j. By definition, there is an e > 0 suchthat

Vfe G IN 3i > k : \\xi-y\\>r + e.

This implies for r' < r + e that e' := r + e — r' > 0 and

VfeGlN 3i>k:\\xi-y\\>r' + e'.

Thus ?/ ̂ LIMr Xi for r' < r + e, which implies ?/ ̂ flr'>r LIMr x .̂ Hence

UMrXi = f)r,>r LIMr'x?.

For r < f, it is clear that

cl ( [j LIMr'x^ = UMrXi = 0.

0<r'<r

Let r = r\ > f and TQ = (f+ri)/2. Since ro > f we can choose a |/o G LIMr°Xi ^ 0.Consider an arbitrary y\ G LIMriXj. Proposition 2.5 yields

yx = (1 - A)2/o + Aj/i G L IM^-^^+^Xi , for A G [0,1].

Consequently,yxe [j UMr'Xi for AG [0,1).

0<r'<r

15

Since\\y\ - 2/i|| = (1 - A)112/0 - Vi\\ -> 0 as A ->• 1,

it follows2/ iGcl( | J

0<r'<r

Hencecl ( [J LIMr'x

0<r'<r

holds true for r > f7 too. •

Let us investigate the semi-continuity of the set-valued mapping

F : IR+ -> 2 X with F(r) := LLVT^.

Due to Bouligand [4] and Kuratowski [6] (compare also [1,2]), F is said to be lowersemi-continuous (l.s.c.) at r, if for each open set U satisfying F(r) l~l U ^ 0 thereexists a neighborhood V(r) such that

£ G V(r) implies F(t) n f/ ̂ 0.

It is called upper semi-continuous (u.s.c.) at r if for each open set U containingF(r) there is a neighborhood V(r) such that

t E V(r) implies F(t) C f/.

We say F is /.s.c. or u.s.c. on I if it has the corresponding property at everyrel.

PROPOSITION 4.4. F is l.s.c. on (f,+00) and u.s.c. on [f,+00).

Proof. Let r > f and U be an open subset satisfying F(r) n £/ ^ 0. SinceF(r) = LlJVr^ is convex (by Proposition 2.5) and int(LIMra:j) ^ 0 (by (4.4)),int(LIMr;Ej) n U is nonempty and open. Therefore, there exists a ball Ba{y)contained in int(LIMr;Ej) n U C LIMrXj. By Proposition 4.1, it follows ?/ GUMr-aXi n C/. Thus (4.1) implies

y G LIMr-CTXi n f / C LIM*Xi n C7 = F(£) n C7 for t e (r - a, +00).

Hence, F is l.s.c. at each r G (f, +00).

Assume now r > f and U is an open subset satisfying F(r) C C7. By (4.1),

F(t) = LWPxi C LIMrXi = F(r) Cf/ for te [0, r].

Consequently, we only have to prove that there is a 5 > 0 such that

F(t) = LIM*^ Cf/ for t e ( r , r + (J).

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If this does not hold true, then there exists a decreasing sequence (i,) such that

Iimtj = r and LIM^x* \ U ^ 0, j = 1, 2,...

Clearly,

t>r JGIN

If y ^ f|i>r LIM*Xj then there exists t > r such that y g LIM*£j. By limtj = rand (4.1), there is atj < t such that ?/ ^ LIM*?Xj, which yields y ^ fl,GiN ^Hence

t>r

Therefore, Proposition 4.3 and LIMra;i C f/ imply

* \ u) = ( P UMtjXi) \U= UMrXi \ U = 0.

But LIM*JXj \U, j = 1,2,..., form a decreasing sequence of nonempty closedsubsets contained in the compact set LIM 1Xj, therefore their intersection is notempty (compare [2, p. 69]). This contradiction completes our proof. •

5. ROUGH CAUCHY SEQUENCES

It is well know that each convergent sequence in a normed space satisfiesCauchy condition, and conversely, each Cauchy sequence in a Banach space con-verges. The relation between roughly convergent sequences and rough Cauchysequences cannot be described so shortly.

Let (xi) be r-convergent, i.e., LIMrXj ^ 0. Take an arbitrary x* G LIMr;Ej.By definition, for all e > 0 there exists an i£ E IN such that i7 j > ie implies\\xi — a;*|| < r + e/2 and \\XJ — x*\\ < r + e/2, which yields immediately

\\xi — Xj\\ < \\xi — a;*11 + \\XJ — a;*|| < 2 r + e.

Hence (xi) is a p-Cauchy sequence with p = 2r. This Cauchy degree cannot begenerally decreased. Indeed, let z E JRn with ||z|| = r and x^ = (—l)lz, then thesequence (x )̂ is r-convergent with 0 E LIMrXj, and p = 2r is its minimal Cauchydegree.

Conversely, let p > 0 be a Cauchy degree of some given sequence (x{). Canwe expect its convergence degree to equal p/2, i.e., LIMp'2Xi ^ 0? Not always.The main purpose of this section is to find a possibly small convergence degree ofgiven p-Cauchy sequences.

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First, let us formulate some properties of Cauchy sequences. Similarly to theboundedness of roughly convergent sequences (Proposition 2.2), the following israther obvious and its proof is therefore omitted.

PROPOSITION 5.1. A sequence (xi) is bounded if and only if there exists anp > 0 such that it is a p- Cauchy sequence.

As (xi) is a bounded sequence in a finite-dimensional normed space, the clusterpoint set C is nonempty and bounded. Thus its diameter D(C) and the radiusR(C) of its circumscribed ball are finite.

Recall that the diameter D(S) and the radius R(S) of the circumscribed ballof a bounded subset S in some normed space (Mn, || • ||) is defined as follows:

D(S) : = s u p \\x-y\\, R(S) := inf s u p \\x - y \ \ . (5.1)

PROPOSITION 5.2. Let C be the cluster point set of the sequence (xi). ThenD(C) is the minimal Cauchy degree and R{C) is the minimal convergence degreef of(xi). That means

D(C) = min{p G H_|_ : (x^ is a p-Cauchy sequence}, (5-2)

and

L1M z. j ^ 0 forr>R(c). [5J>

Proof, (a) If 0 < p < D(C) then for e = (D(C) - p)/3 there exist two clusterpoints c\ and c<i such that \\c\ — C21| > p + 2e. For all k G IN there are ii, %i > ksuch that

ll^ii — ci|| < e/2 and ||XJ2

which imply

> ||ci - c 2 | | - U l a ^ - c i H + Ha^-c 2 | | )> p + 2e-(e/2 + e/2)

= p + e.

That means (x^ is not a p-Cauchy sequence if 0 < p < D(C).

Assume now p > D(C) and e > 0 is arbitrary. Clearly, there are only finiteXi outside C + B£/2(0), otherwise there exists a cluster point outside C + B£/2(0),because the space considered is finite-dimensional and C + B£/2(0) is open. There-fore, there is an i£ such that Xi G C + B£/2(0) for i > i£. That means if i\ > i£

and i2 > i£ then \\x^ — ci|| < e/2 and ||XJ2 — c2\\ < e/2 for some ci, c2 G C, whichyields by ||ci - c2|| < D(C) < p

< p + e/2 + e/2

= p + e.

Hence, (xi) is a p-Cauchy sequence if p > D(C). Thus (5.2) is proved.

(b) If r < R(C), by (5.1), for all z e JRn there exists y G C such that\\z — y\\ > r, which implies by Proposition 3.5 that z (£ LIMrXj, i.e., LIMrXj = 0.

If r > R(C), by (5.1), there is an x* G JRn such that ||x* - y\\ < r for ally G C, which yields x* G PLec Br(y). Hence, by Proposition 3.5, LIMrXj ^ 0.

It follows now from (4.2) that R(C) = f. Therefore, by Proposition 4.2,LIMrXi ^ 0 for r = f = i?(C). Thus (5.3) is completely proved. •

The latter proposition shows that R(C) is the minimal convergence degreeand D(C) is the minimal Cauchy degree of (x{). To find the relation betweenconvergence degree and Cauchy degree of a sequence, this result gives us a hint touse the relation between the radius of the circumscribed ball of a bounded set andits diameter, which is given in the following.

PROPOSITION 5.3. (H. E. W. Jung [5], see also [8, p. 78]) For any boundedclosed subset S of the Euclidean space (H n , | | . | |) ; the following inequality holdsbetween the diameter D(S) of S and the radius R(S) of the circumscribed ball:

The inequality (5.4) becomes an equality if and only if conv S contains a regularsimplex with side length D(S).

PROPOSITION 5.4. (Bohnenblust [3]) Let S be a bounded closed subset ofsome normed space (Mn, ||.| |). Then the following inequality holds between thediameter D(S) of S and the radius R(S) of the circumscribed ball:

Note that Bohnenblust formulated this result in Minkowski geometry. Leicht-weiss [7] showed later that n/(n +1) is the best factor by showing cases where theequality holds true.

Let us state a relation between Cauchy degree and convergence degree.

PROPOSITION 5.5. Let (xi) be a p-Cauchy sequence in some normed space(IRn, || • ||) for some p > 0. Then it is r-convergent for r > -^h[P- In particular, if

|| • || is the Euclidean norm, then (x^) is r-convergent for r >

Proof. Since (x^) is a p-Cauchy sequence, by Proposition 5.2, we have p >D(C), where C is the cluster point set of (x^). Therefore, by Proposition 5.4

TI rir > -p implies r > ~D{C) > R{C).

Consequently, by Proposition 5.2 again, LIMrXj ^ 0 if r >

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In particular, if || • || is the Euclidean norm, then we obtain the remainingassertion by using Proposition 5.3 instead of Proposition 5.4. •

Remark. The parameter n given in Proposition 5.5 is the dimension of thenormed space considered. We can obtain a better result by replacing n throughthe dimension m of the cluster point set C of (x^ (i.e., of the affine hull of C),because m is often smaller than n, and in this case we have

n m n I m> —7 and , / ———— >n

To conclude this section, we consider the rough limit in some proper subsetsof TRn of p-Cauchy sequences.

If (x{) is a p-Cauchy sequence, then for all r > p there exists a k E IN suchthat

{Xi : i > k} C LlM(x^'rXi. (5.5)

In particular, for (x^ as a Cauchy sequence in the classical sense (i.e., p = 0) andfor all r > 0, there exists a k E IN such that (5.5) holds. Of course, if (x^) is asequence in some subset M, then LIM^^^Xi C LIMM'rXi implies that

UMM>rXi ^ 0 for r> p,

although M is possibly not closed. This property is also available for smaller r.The following gives an example in Euclidean spaces.

PROPOSITION 5.6. Let M be a convex (and not necessarily closed) subset ofthe Euclidean space (H n , || • ||), and (xi) be a p-Cauchy sequence in M (for instanceM = conv{xi : i E IN},). Then

Proof. Consider the cluster point set C of (xi). Clearly, both C and itsconvex hull conv C are closed and bounded. It is well known that C has a centerz E IRn of its circumscribed ball, i.e.,

s u p \\z — y\\ = R ( C ) = i n f s u p | | x — y \ \ ,yec xeiR" yeC

which is unique and belongs to convC (compare [8, p. 78]). It follows C CBR(C)(Z), which yields by (3.3) that z E LlMR^Xi. Moreover, since C is thecluster point set of (x^ C M, we have C C clM. Therefore, z E c lM follows fromthe convexity of M and z E conv C.

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If r > R(C) then, for a = r - R(C) > 0,

Ba{z) C LIMR(c)+aXi =

Thus z E cl M yields

| / M n Bff(z) C M n LIM r^

If R(C) > r > . 2(n+i) P then Proposition 5.2 and Proposition 5.3 imply

Tt 71

Consequently,

VBy Proposition 5.3, this equality is only fulfilled if C contains {ci, C2,..., cn+i}with ||CJ — Cfc|| = -D(C) for j ^ A;. Since z is the center of the circumscribed ballof the regular simplex conv{ci, C2, • • •, cn_|_i}, we have

, n+i

~T~int(conv{ci,C2,.. .,

Consequently, since {ci, C2,..., cn+i} C C C c l M and M is convex, z must belongto M. Therefore, we finally have

z e M n LIMfl(c)x?- = M n LIMrx7- = LIMM'rx,-.

6. CONCLUDING REMARKS

Although it is assumed that the normed space considered is finite-dimensional,some of the results presented in this paper remains true in infinite-dimensionalspaces, for instance Propositions 2.1-2.4. Some other conclusions are also valid ininfinite-dimensional spaces under certain compactness assumption.

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ACKNOWLEDGMENTS

This paper was written while the author was visiting the Abdus Salam Inter-national Centre for Theoretical Physics, Trieste, Italy, as an Associate. A generousgrant from the Swedish International Development Cooperation Agency facilitatedthe visit. The author would like to express his sincere gratitude to both bodies.

REFERENCES

1. J.-P. AUBIN AND H. FRANKOWSKA, "Set-Valued Analysis," Birkhauser, Bo-ston Basel Berlin, 1990.

2. C. BERGE, "Topological Spaces," Oliver & Boyd, Edinburgh and London,1963.

3. F. BOHNENBLUST, Convex regions and projections in Minkowski spaces, Ann.of Math. 39 (1938), 301-308.

4. G. BOULIGAND, Sur la Semi-Continuite d'lnclusions et Quelques Sujets Con-nexes, Ens. Math. 31 (1932), 14-22.

5. H. E. W. JUNG, Uber die kleinste Kugel, die eine raumliche Figur einschliefit,J. Reine Angew. Math. 123 (1901), 241-257.

6. K. KURATOWSKI, Les Fonctions Semi-Continues dans l'Espace des EnsemblesFermes, Fund. Math. 18 (1932), 148-159.

7. K. LEICHTWEISS, Zwei Extremalprobleme der Minkowski-Geometrie, Math.Z. 62 (1955) 37-49.

8. K. LEICHTWEISS, "Konvexe Mengen", Springer-Verlag, Berlin HeidelbergNew York, 1980.

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