View of On the behaviour of the tangential modulus of a Banach space I

REVUE D'ANAI,YSE, M;NNÉNTQUE ET DE rrruONrE DE L'APPRO)ilMATION

Tome 24, No' 1-2, 1995, pp. Z4l-248

ON THE BEFIAVIOUR OF THE TANGENTIAL MODULUS

OF A BANACH SPACE I

roAN _$ERB

(Cluj-Napoca)

l.INTRODUCTION AND NOTATION

In some investigations on Banach spaces and their applications

it

is some.

times useful to know the geometry of the unitballs. The geometry of theballs rnay be reflected in the behaviour of som.e moduli, i.e. of sorne real functions attacheâ to a Banach space,

In this paper the properties of such a modulus are discussed. The invoked modulus lias been recently introduced and used (see[7]) at existence problems for

the Lipschitz continuous selections of set-valued mappings. A new geòmetric defi- nition of this modulus is given. For sorne reasons (see Proposition 2.6)

it will

^be

called the tangential modulus.

From the behaviou¡ of the tangential rnodulus in the neighbourhood of some points we obtain information about the geometry of the Banach spaces, A charac- tenzation of the unifonn convexity of a Banach space is reconsidered. The con- vexity of the tangential modulus in the neighbourhood

of I

arrd corurections with known moduli is presented too.

Let (X, _ll' lll ^{Ut a} real Banach space at'ñ, let

X*be

its dual. To avoid

triviali-

ties we assume

thatxhas

dimension at least two, For

r

> 0 and _r e xdenote _by B(x,r) the closed ball with center -x and radius

r

and by B(x)

:

_B(x,ll ^,

ll) ^ure unit ball

ofx

Analogously,

^s(,Y)

will

represent the unit sphere

ofx

choos ing x,

y

e

x,

x + y we shall consider the straight line passing through

x

and

y

clenoted by xy as

well as the open and the closed line segment with the vertices

x

and

y

denoted by (x;

y)

respectively by _fx; y). Let

\y,z

^e

X

^{x +}

^y.A

^{parallel to}

ry

from z is the set {z + }' (x

-

^{y) : À e R}, The symbol}

rry will

be used for BirkhofÊJames's orthogo-

nalifyi"Ø,ll'll);"a-'ty:xLp¡lif llÌll ^< ll'*

^Àyll

^{, forallÀe}

^R.

The modulus of smoothness

ofXis

defined [5] ^by:

p*(')= p(,) ='*{;fl x+ryll*ll"-

"yll-2). x,! ^es(x)}, t)

⁰

and the modulus of convexity [5] by:

ô,(r)

=

i,'r{r

_t

-}n.+yll: ^x,.y

^e

_^s(x), ll"-rll

⁼

_"} ^,0<eaz.

The Banach spacexis said to be uniformly cot'Nexif

ðle)

> 0, 0 < ¿

12,

_artd uníformly sntooth

if 1im.-6 px þ)

^I

r

₌ 0.

Tbe ortltogonal modulus of srnoothness (see T. Figier [3], p, 129) is the flinc-

tion Þx

^{defined by}

Þ,(,) =.po{;fl x+tyll*ll" -,yll-2): ^x,v

^e

s(x),

and

there exists ,r* e

^S(X*) ^{such that} ;*(r)= l, x*(y)

=

O] , ^,

^{> 0.}

It is well known (see D. Arnir l1l, ^{p. 33) that the} condition of orthogonality used

in

the definition

of þ¡ ^is

equivalent to BirkhofÊJames's orthogonality.

Forany x,y

e Xwitft llyil. l.ll.rll,

^{thereis aunique}

z:z(x¡)tn(x;1) witti

_ll

"ll : _t.

_{We put as}

_iri'[7]'

ot(x,y¡ = \ _'¿ t ll'll-r ll";'Í"'Y)il

_.^'

and define the function

E: E*:

^[0,

])

-+ /? by

€(p)

:sup{' (*,y)'llyll=B <1< ll"ll }, ^{0<B <r,}

K. Przeslawski and D. Yost name this function in

t'o

different ways (in the two variants of the preprint [7]).

242 Ioan _$erb

2. PRELIMINARY RESULTS

,.

^In the sequel the follorving sirnple geometrical lemrna

will

be freque¡tly applied,

L¡waZ,t.If xI3¡/ ^and 01a<b,

then

llx

⁺

ayll .ll, * tyll

.

Proof. From the BirkhofÊ James's orthogonalify we have

ll

"

^{ll all}

x +

ayll .

If

_il

'll ⁼

^ll_l

^{* oyll}

^{tnen x +ay}

^rB¡r

^{aud rhe result foilows.}

^rr

^'jl

^,

^il

.

_ll

, ^*'rj,¡¡

tlrentlrecollinearpoints.r,

x*ay,r+b\,0<a<b

areinthisorderintheinterioi, on the bounclary, respectively in the exterior

of

B (0 _{, ll} x + oy

ll

) ^o

Tangential Modulus 243

One observes that symmetrically

if x Iily

and.

b < a < 0

then also

ll, ⁺ _"yll . _ll' * uyll

^,

The following useful result can be find

in

[2], [8], [4].

Lpvvn

^{2,2, Let}

X

be

a

two-dimensional Banach space and let K1,

K,

be clo s ed co nvex subs ets of X u,ith nonvotd inter io rs.

If

K _r c. K r ^{t h en} r ^(K

) ^<

^{r ( K 2),}

where

r (K¡)

denotes the length of the círcumference of

K, i:

1,2, Now, we present a first result with respect to the function

(

^.

L¡vva

^{2.3. Let}

^X

^{be a} two-ditnøtsional Banach space and

t

^{y e}

^X

^{be srch that}

llyll=p<r.ll'll.

Then tltere exísts a vector

y'e B(0,

p) such that

xy'

^{ís the} supporting line

of

B(0,

B)

^and

a(x, y) <

r¡(¡ y'),

Proof.

If.r

andy ^are linearly dependent, then

o

^(x,

y'

) > c¡ (¿

y) = l,

^for

every

y'e

B(0, _P). In the other case,

in

every semi-plane detennined by 0x there

existsasupportinglineof B(0, B)passingthrough

x.Letxy',þ,,'e

B(0, p)) _bethe supporting line of .B(0, B) contained

in

the semi-plane determined

by

0:r and y.

Therr the triangle with vertices 0, Jr, z(x, y) is contained in the triangle determined by 0,.r, z(x,

y').

From Lemma 2,2 wehave:

ll'þ,v)

^ll

.

_ll ^x

_- z(x,r)

_ll +

ll

'

^{¡¡ =}

=ll'G, ^r')

^{ll +}

^ll, - "(,,

^v')

^{ll. ll'} ll,

and hence a(x, y)

l r(¿ y'), a

Remark 2,4 Frorn Lemma 2.3 the function

(

may be defined by

((B)

₌ sup{r,r

(r,y),llyll= F,ll'll> t, yLu(x- y)} ,

^{p e}

fo,r)

^.

Levva

2,5.

Let X

be a huo-dimensíonal Banach space and let

xy

be in

X

suclt ^thctt

llyll.p.r.ll'll,

If x' e

(x; z ^(x, y)) then

a(x, y) 3 a(x', y).

Proof. We have a(x, y) = ^cù(ï',

y): I

^for x andy two collinear vectors. Let-r-

andybe linearlyindependent. Denotebyx, theprojection

x lllxll of;

onB(X);

analogously ^x't

= I ^f

_ll

f ll

It is clear that z(x, y)

:

_z(x',

_y):

_{z, The parallel to}

2 ³

the straight hne xy from origin intersects the straight lines zxr, respectively zx,

, itt

-r, respectively x'r.

A

comparison of similar triangles

xrrx

andr,rrO

yi.tO,

^¡

ll,- "(,,ùll- þ.1

ll"ll-ll",llll",ll

and tlren o>(x,

y) : _llt ll

^{By a} similar argument _one obtains ot(x,,

¡¡ :

ll

";

^ll

From the convexity of B(X)

it

foilows thar

ll xr

ll

= ^ll

¿ ll ^o

PRoposlrrox 2,6. Thefunction

(

can be defined by

€(B)

='up{ _ll" -"ll:,

^u

_^S(x), ^v

ç

x,

^u

^Lorv,

(l)

'rufll ^(t

- ^{),)u+ I'v}

_ll ^{= p}

_J,

for

eaclt Þ

e

_[0,

]),

natnely

\(p)

represents the maxinrul length of the line seg- ments

lu;trl,

^{of the} batt B(O,p) whitett e

S@

attcl

uL,,v.

Proof.

is sufficienr to suppose (in or (.r;

y)),

_ll

yll"":

^p

and that.ry

i

e ball B(O,p). Then using Lemma 2.5 ancl the corresponding notation one gets:

6 (p) <

'rp

^{{ ll}

x,ll:

" (* _,

y) _{ro, (,} (,

_, y) ^_

*,)

,

(*)

yror(! -'(r,/))],

^o

^<B

^{< 1 .}

Lr fact

in

(x) instead of an inequalify we have an equality as we ca¡ see a little later. Now.

if

the parallel to the support line

xç

of B(X) from origin iltersects thc straiglrt line ;:y

in

v and

if

u'e rvrite u

: _z(r,y),

_{then from the} parallelogra n-t

tr\ûx, lvelrave

_{11.", ll =}

ll"-uii, llrll ⁼ r, ulwv, min,.,oll(t_rl _{u_t)uvil=}

_B

,

^u,r¿

fonrrula (1) followq, It rernains to prove the reverse inequalify i,,

(*).

Let zbe in

^s(-ll and

d

be asupporti'g

li'c ofB(.!

passing trrrough z. supposoy e B(0, p) _and zy is a support line ofB(O, p). Denote rty zrthe vector

(l

+

r)

^{z-wttl.te >} 0 fìxed and by zrthe intersection of ^{(y; z}

)

^rvith ^SQf . In the trvo-dirnensional space spa¡necl by..

y

and z the parallel to yzr(r'espectively

to

z) fi'om origirr intersects _the straiglrt line z:r(respectívely d) i'-r'r(respectivery-rr).

The'r¡ (zt,y)= iltrll Ife\0irrni,

tends

to a

vector

-r", coilinear to x, By

frre

"onvexiiy k ⁿ¹⁴ we

rravá

il"",ll ^> il¿ ll ^o

TIte new defirrition of

(r

^enables us to call now this functiolrthe tøtrgenti1l tnodalus of

X In [7]

^the authors have presented solne applications ancl esse¡tial properties of the tangential modulus. Forinstance,

it

was proverl that

( is

an iucreasing fi.urction, 6(0) ^=1,

6"(p)

^<

(r*p)/(r-B)

₌

_6,,14(F),

_B ^e

_[o,r)

and that

if flis

a Hilbert space then

€H(p): (l -

^{pz¡-u. The} locally Lipschitz conti- nuify

of (

was obtained by the sha¡p inequality

1,(v) - 6"(B)

^<

€,'r,l(v) -

€,

'14(P)

=

^,

0 < p ^< y < 1. The geometry of the unit ball ofXwas reflected in the behaviour

of

the function

(.

For instance, one obüains that X is uniformly convex

if

and only

if

lim

inf

_pzr(1

-

^{P)6r,(P) = 0.}

Tangential Modulus

3, TIIE BEHAVIOI]R OF TTIE TANGENTIAL MODULUS

IN THE NEIGIIBOI]RHOOD OF I

244 Ioan $erb _{4 5} 24s

In the sequel we sliall continue the investigation of the properties of

(

insist- ing on the relations between the behaviour of the function

(

and the geometry

of

the unit sphere

ofX

First of all, one observes that

lirnp-,ËlF) :

oo. For the proof it is sufficient to consider only the two-dirnensional spaces . Let F be a two-dimensional space arid let ube a point of smoothness of the unit sphere

^9(,F). Denoteby d ^the support line

of.B(fl

^{passing through} uandby draparallelto dfrom origin, ^Choose vne d, such tlrat _ll

,,

_ll

:

_{n e N . Then the straight line} uv,contains at leasta point t, with llq,ll

:

_F,

.

1, We ^have that

(lp,) )

^r¡ and since

(*is

increasing

it

follows that

timp-r€lP):

^co.

On the other hand

if

^p is chosen so closed to

I

that

EIB) >

^{1 +} B and

if

tlie

vectors u ^{and v} in formula (1) verify ll"-"ll t ¹ f ^p then min2r¡

ll(t - tl

^{u + )uv} ll ir attai"eaforl. e (0,1).Indeed', fo.l.'^S1O,F)and

t:?,.v4Q

_-?ù;;,

rvithÀ) l weliave

ll"-

^ull ₌

il,-

^{úll <}

^{ll,ll *llr}

^ll

⁼ t*8,

whichis irnpossible. It is clear

that

€,,trl(F) >

1+Þ, forall

_P

e

(0,1) and so in tlris case À

in

forrnula (1) ^can be taken

in [0,i],

^{In the opposite case when}

Xis

^a

Hilbert space we have that if I e .S(O,p) ancl ^¡

I

^(u

_-

^{t) then from the} orthogonalify's symmetry

it follows: ll" - ^tll . _{llrll =} t .

Since

{lp) )

1, for

all p e

(0,1)

it

irnplies that À in fonnula ⁽¹⁾ can be taken in [0,1], I ^{do not} know

if

^1" in fonnula ⁽¹⁾ can be taken only in _{[0,1] for} every Bariach space Xand every p

e

^(0,1).

IJow we cornpute again the tangential modulus €nG) ^{whe¡e I-I is a} Hilbert

space and

('

_| ') denotes its inner product. In this case we have:

246 Ioau $erb _6,

7 Tangential Modulus 24',7

ll" - *,ll

₌

_ll, - ^u,

_ll ^,

and

,[ry) =+ll"-uu ^ll* ll,-u,lD*"

₌

jlerol*E(v))+e _,

^e > 0 and the convexity of

{

follows. Ë

As it is well known (see V.I. Liokoumovich [6]) ^the modulus of convexity ô, is not always a convex fuirction, but it is a sirnple exercise to prove the convexiú of modulus

of

smoothness p¡ and the convexity

of

the orthogonal lnodulus

of

smoothness

p" . ^Now,

because'the

tangential modulus is convex in

^a

neighbourhood

of

1 we can expect that there exists a strong relation between

("

and

po

(respectively

px .)

Such a subject

will

be treated elsewhere in the second

part of this

paper. There, the behaviour

of

the tangentiai modulus

in

the neighbourhood of 0 is crucial. So, it is natural to study also the behaviour of

("in

the neighbourhood of

L

In this direction the following proposition was provecl in the second variant of the preprint

Ul,

In the spirit of this paper we give finally a new

proofofthe "if"

^part.

PRopostttoN 3.2. The Banach spqce

X

is uniformly convex

if

and only

if

rim

^infpzr (r

- ^p)E"

(p) = o ^.

P,oof. Suppose

thatxis

unifomrly convex

and lirn in$rr(1-

_{F)Ex (F)} = ^{o .}

Letu e S(X),v eX

^{be suchtliat}

uLurvand mirl^rolli"u+(t-f)rll=Þ . f.t

y

e uv

n

^{B(O,p) and let w be the seconcl} intersection of

uvwíthB(,f].

The segment line [0; ø] intersects B(0, ^p) in ø,. In the two-dirnensional space generated by.ø and v the parallel to 0 y from u, intersects uv in vr.

It

is clear that v,

e

_ly; u). A compari- son of

sinilar

triangles uurv, and u.v yields:

; ^{with î/,v} efl such that

u e

S(ø),

""0

^:f;:lllx"

+ (r

- ^r)u

_ll = P

It means that for Hilbert spaces the ,,sup"

in formula

(l)

can be omitted. Indeed, let u,vbe as above. Then

^ï,'llll

xu +

(t- r)

u

ll'

⁼

=

^l¡å1,1[^'

(r ⁺ _ll

v

_ll')

- ^zx¡" _il' ^*

il

"l'f= ^il,ll't

^{(r + ii}

^{v ll')}

^{= ø,}

we have _ll

" ii: ^pe -

^{p,)-% and,}

ll" - ,ll

^{= (r}

* llril')"'

^{= (r}

- ^p:¡-rr'

= Eu

(þ)

^,

PRopostrtoN 3.

l.

^{h-or every} Banach space

x

^the tangenríar ntocrurus E,r

í,

o cotrexfuncríon ín a neigltborhood of L

Proof. From the

co'tinuity

_{of {} it is sufficient to prove that:

€.'(P) = (u

l"- ^rll

lu) =o

).+fr(p)*E(y)), ^p,

^{< F}

^<y

^<

^r,

þ:41

yliirr

^{p6 is' chosen so that} €(pd ^>

2>

r + ^po. Let u.v

e

xbesucri rrrar

\((p

^{+ y)rz)}

s

_ll¿r

- ^{yll + s. e >} 0 being arbitrarity small. Here

_lj

"ll : _{t, ulo,}

_v,

rnirrrno,,,ll

xu+(t-1")vll=(p+y¡t z=l!/1, y

beins

in

fu,vlarrcl ^po

< ^{p <}

y

< _l.

äîi:;äïiî:[ïJ;:îiliï',::,::::^,f ,:,?,;JÁ',?;:i!:,.î{î,!

:

_uyT

_n

_{0v. The}

orritl"]

^{to 0y from v,} intersects øv respectiv ery

,t,.,in

z resl-rcc_

,,u.",ry.traud the_¡rarallel

to

uvfrotnzriniemects 0vinruu.

It

^js cleartrlatz

js

_trrc nriddle

poi't

_{of IvO;zrl and}

_vis

then.í¿¿1"

poi.t

of ¡rO;rlrrJ. We have :

e[]).rr

^u

- vl+e-ll=l *ll."

=

. )(ll" -,,11*ll" -,,

_|i)

^+,

Since wre

I

^{v,' vrJ and}

ulo,

v,by Lernma 2.1 one obtains:

ll,ll -ll,-"'ll ll,-"ll

(r

-

B)ll

" ^-,,11= ll, - _o

_ll ^.

Passing

to

supremurn over

all

pairs

(u, v) with _llr ll : _{l, uLu, y}

_and

rnin^,'

_ll ^1,"

*

(t

-

^l,)

_"ll

⁼

þ

^{we have}

(r- ^p)€(p)

=

.upll,-

", ^ll

.'upll ø-,ll

^.

For every p sufficiently close to 1, there exists a pair (ø,v)

such that

ll"- *ll,

^t,

tz ,ll"ll =llrll = t

un¿from

ylrr(u-w) itfollows:

248 Ioan $erb 8

'-llryll='-llvll ^=r-P

RE\¡IE D'ANAr,ysE

m¡uÉnreun

ET DE

rrnonm

DE L'AppRo)ilMATIoN Tome 24, No. 1-2, 1995, pp. 249-250

and so

Ex(blz):

^{0, â} contradiction with the unifonn convexify of X,

t:

BOOK REVIEWS

REFERENCES

1. Arnir, D., Characterizations of Inner Product Spaces, Birkhäuser Verlag, Ba-se1-Boston-Stuttga4 1986.

2. Busernann, H., Tlte Geometry of Geodesics, Acadcrlic Press, New York 1955.

3, Figiel, T., On the ntoduli of convexity ancl sntoothness', Studia Math,, LVI (1976) 121-155.

4. Ka-Sing Lau, Ji. Gan,On two class'es of Banach spaces wilh uniþrm nortnal strucÍure, Srtt(7ra Matli., XCIX,

I

(1991) 41-56,

5. Linderrstrauss, J., Tzafriri, L., Clas,sical BanacJt Spaces II, Function Spcces,, Nerv York 1979, 6. Liokonmoviclr, V.I., Tlte eústence of Banach spaces'with non-conyex nodulus' of convetit.¡,

(Russian), Izv. Vysir. Ucebn. Zaved., Mathematica 12 (1973) 43-50.

7. Przeslawski, K,, Yost, D., Lipschitz selectíons retractiotls ancl eúensiotts, Preprint (preliminary version) 1992 ard 1993 variant.

8. Schäffèr, 1.J., Geotnehy of Spheres ín Nornterl Spoces, Dekke¡, 1976.

MARIO MILM Al'1, Extrapolatíon and Optírnal Decontposítions, (withApplications to.4.nalysís), Lechne Notes in Malhemdics, vol. 1580, Springer-verlag, Berlin-Heidelbøg-Nerv york _1994,60pp.

Received 15 IX1994 Dep arhnent of Ma.Íh enntics

" B ab E-B ^ob,aí

"

Universíty

3 400 Cluj-Nopoco, Rontânio

The extrapolation theory, elaborated by the author, mainly in collaboration with B. Jawerth (see Bjöm Jawertb and Ma¡io Mihnan, Extrapolation Theory rvith Applications, Mernoirs Amer.

Math. Soc. vol. 440 (1991) is coricemed with aproblem which is sourehol converse ro the interpo- lation problem.

Tlre basic notion of iulerpolation theory is that of Banach couple, whicb means apair

A :

(Ao, A

)

^{of Banach spaces enrbeded in a} Hausdorff topological vector space 1L Let A (A ₎ =

Aol

^A, and

D (A-) :

Ao ⁺

A,

equipped

with

the norms

ll'll^

⁼

'nax{ll'll,:i

⁼

o,r}

and

ll'll,

^{= inf}

_{ll'.lL, ^*ll¿

llr,

''

^{= xo #x7, x,}

^eA¡,

^{t =0,1}}

^,

respectively. An intennediate space is aBanach spaceA suchtlratÁ(7 )-+-,i-+

X(l

_). The spacesl andB a¡ecaiied interpolation spaceswithrespecttothecouples Á:(Ao,A) and B- = (Bo,B)if

r: A

_{-+ B} implies T:A-+8.

If moreove¡

ll,

llr."

,

'o"* ^(ll

r

_{llr. ,o , ll}

,

llr,.",) ^{tben A, B are caled exact} interpotation spaces. An interpolation rnethod is a functor ,l? defined on the category of Banach couples and linea¡ bounded operators benvecn tlrem, such rhat F(A _), F(E) are iuteqpolation spaces for A ,

E

^and F(D = T for

all T:

7

^-+ ,B . The interpolation method is called exact if it yields exact interpolation spaces. A good reference for the interpolation theory, both classical and abstract, is Yu. A, Brudnyi and N. ^ya, Krugljak, Interpolation Functors aud Interpolation Spaces, vol. I, North-Holland Math. Library vol.

4:7,718 pp., Anrsterdam New York, Oxford. Tokyo, 1991.

The extrapolation theory is dealing rvith the converse problem: Given a faurily of interpola- tion spaces reconstructthe originating pair. In this fonnulation, the problem is directly related to best possible interpolation theorelns and in some sense, it could be considered as a chapter of interpola- tion theory of infinitely many spaces. The precise connection betleen these theo¡ies is a¡ ^open

problem.

The book is dealing also rvith weaker fonnulations of the problem, such as the extrapolation ofthe continuity ofan operator l"or the extrapolation ofinequalities for its nonn, usually based oti the basic fuuctionals K and J.

More exactly, let {Ar:0 ^{e @} be a} family of Banach spaces indexed by some fixed index set

@ (usually @

:

(0,1). These families of Ba¡rach spaces are strongly compatible in the seuse that there are two Banach spaces Á and Ð (depending on the family Øu)) ^{such that L} c ArcL,O e @. If

ft.)

and {8.} ^are two families of shongly compatible Banach spaces, a, c ArcE", auc Brcl,o, a natu¡al morphism is a bounded linear operator T : _{Ar)

+

_{8.}, ^i.e. T :

),

-+ E, is an operator whose reshictions to

l,

maps

l.

^into B, with nonn

I

l, 0 e ^@.