A N N A L ES S O C IE T A T IS M A T H E M A T IC A E P O L O N A E Series I : C O M M E N TA TIO N E S M A T H E M A T IC A E X X I I I (1983) R O C Z N IK I P O L S K IE G O T O W A R Z Y S T W A M A TEM A TY C ZN EG O
Séria I : P R A C E M A T E M AT Y C Z N E X X I I I (1983)
Б. So l t y s (Poznan)
Coinplementably universal spaces for p-Banach spaces oî finite dimension
Abstract. A construction is given of a separablep-Banach space which is (isometric- ally) complementably universal for all m-dimensional (resp., all finite-dimensional) p-Banach spaces (0 < p < 1).
Let 0 < p < 1.
A p-normed space is a pair (X, v), where A is a linear space and v is ър-погт on X , i.e., a function v: X ->jB+ such that v{x + y) < v{x)-\-v{y)1 v{tx) = \t\pv(x) and v(x) = 0 only if x = 0, for all x, у in X and all scalars t.
(The field of scalars is either R or C.) A p-normed space (X, v) is always considered with the metric d(x, y) = v(x — y) which converts it into a metric linear space. If this metric space is complete, then (X, v) is called a p-Banach space.
A p-normed space X — (X, v) is said to he (isometrically) complement
ably universal for a class S of p-normed spaces if for every X — (X, w) e ê there is a linear isometric embedding j:X -> X such that there exists a pro
jection P of X onto j(X ) with ||P||„ = 1, where ||P||„ = sup^Pa?): v(x)
< 1}.
We slightly modify the construction of A. Szankowski [6] for the case p = 1 and q = 2 so as to obtain the following two results.
Th e o r e m. For every p e (0,1], q e (1, oo) and m e N there exists a p-Ba- nach space which is isomorphic to the Banach space lq and is com
plementably universal for the class m o f all m-dimensional p-Banach spaces.
Moreover, i f p = 1 , then the same is true also for q = 1 .
Co r o l l a r y. There exists a separable p-Banach space with separating dual which is complementably universal for the class p o f all finite-dimen
sional p-Banach spaces.
It is enough to define 9Cp as the 7,.-product of the spaces %рлт provided by the Theorem,
oo
= ( © %рЛ,ш)г1 where К r < oo.
m = l
134 B. S o ï t v s
Recall that if (Y n, vn) is a sequence of p-Banach spaces and r > 1,
CO
then {® (¥ n,v n))r denotes the p-Banach space consisting of all sequences
n — 1 oo
{Уп) such that Уп e Y n and v({yn)) = ( 2 v rn(ynj)llr < oo, under the p-norm v
n = 1
given by this equality.
R e m a rk 1 . For 0 < p < 1, if a p-normed space X = {X, v) is such that for every m there is a linear isometry j m :1™->X, then X is not isomor
phic to a normed space.
In fact, otherwise B = conv {x e X : v(x) < 1} would be a bounded subset of X. However, this is impossible, because for xm = j m{ljm , ..., 1 jm) we have xm e B and v(xm) = ml~p-+oo as m->oo.
Thus, in particular, a p-normed space complementably universal for $Fp cannot be isomorphic to a normed space.
R e m a rk 2. Let (X, v) be a p-normed space complementably univer
sal for ^ p>m, and suppose it has a separating dual. The latter assumption implies that if w is the Minkowski functional of W = conv V, where V
~ {x e X : v(x) < 1}, then w is a norm on X and W is its open unit ball.
We wish to observe that the normed space (X, w) is complementably uni
versal for i.e. for all m-dimensional Banach spaces. (The same holds if Ppjm, and Ж1>т are replaced by and & x.)
Indeed, let (E, Ц-Ц) e #"1>то. By assumption, we can find an isometry j: (E, ||*1П->(Х, t>) and a projection P from X onto F = j(E ) with ||P||„
= 1 . Hence P (F ) = V n F = j ( { e e E : Щ < 1}) and so V n F is convex.
Therefore, P(TF) = V n F c W n F c P ( W ) , whence (w]F)p = v\F and l\Pj\w = 1. Thus j is also a required isometric embedding of (E, |j-|]) into (X, w).
Our observation and a result of Bessaga [1] now show that if m > 2, then dim X = oo.
Similarly, using a result of Johnson and Szankowski [2], we see that no separable p -Banach space with a separating dual can be complement
ably universal for all separable p-Banach spaces with separating duals.
(For stronger results see Kalton [3], Theorem 6.8 and Corollary 6.9.) P ro o f of th e T heorem . Fix an w-dimensional linear space X equipped with its unique linear Hausdorff topology. Denote by J f p = X ^ X J-th e set of all p-norms on X ; then every F e ïF p>m is isometric to (X, v) for some v e J f p. Let Жp = Жp (X) denote the set of all continuous functions w: X -> R + such that
(a) w(tx) — \t\pw{x) fo r all x e X and all scalars Ц
(b) there is v e JTP fo r which v < w < 2v (hence w(x) = 0 iff x — 0 ).
For wx, wz e Жр we shall write wx < w% if wx(x) < wz(x) for all 0 Ф x e X.
Gomplementably universal spaces 135
Let и be a fixed p-norm on X.
We start by choosing a sequence (wf) which is order-dense in <^,i.e. has the property:
i f w w " e Жр and го < w", then w' < wi < w" for some i.
This is possible by the following argument: Жр may be identified via the map w->w\s with some set of strictly positive functions in the Banach space G(S), where 8 is the unit sphere for u. Since G (/S') is metriz- able and separable, so is 0>1 and hence there is a sequence (wf) in Жр such that the restrictions w{\s form a metrically dense subset of Xow if w', w" e Жр and w' < w", then
a — min (w" (x) — w1 (so)) > 0 ,
xeS
w[ Ar\au e Жр, and there is i such that
max|u/(&) -fa /2 — го^х)] < a / 3 .
xsS
Then w < w i < w " .
Using condition (b), we choose also a sequence (v{) in J f p such that (1) vi < wi < 2vt, i e N.
oo
(i) F or every v e J f p, r > 1 and (an) e lr with an > 0 and arn = 1,
n ~ \
there exists a strictly increasing sequence (in) in N such that OO
(2) vr(x) — y j win(x) f or c d lx e X )
n = l
(3) a tv < wix < v, \anv < win < anv for n > 2.
To prove this we proceed as in the proof of the Lemma in [6], re
placing therein the exponent 2 by r and choosing a > 0 so that (1 -f a)llrsn < an+1v and aarn+1 < X + 2-
Bor the rest of the proof define r and s by
g , 1 1
r ~ — > 1 and — 1— = 1 ,
p r s
and fix 0 < a < 1 satisfying a certain additional condition which is to be specified below in (8).
To every v e JT p with v > и we assign a sequence (an(v)) as follows:
we first choose a — a(v) so that
(4) max ((1 — a)llr, b, c, d) < a < 1,
136 B. S o l t y s
where
Ъ sup
u(x) — \
u(x)
v(x) 1, sup
u(x) = ]
L i « ( e ) \ y I U(®)// < i , d = (V 3 - l ) llr> i ,
and then define an — an(v) by
ax = a, an = (2*31-n(l — ar))1/r for w > 2.
Note that (5)
n = l n= 2 n = k
(6) (7)
< и < a-fl for n > 2 ;
oo
n = l
3 * -1 + 2 *
3 * - l
where t = (q — l)j(r — 1). The additional requirement on a which is needed in part (iv) of the present proof is this:
(8 ) а < ( 8 0 Г г.
With the p-norm v and the sequence (an) just defined we now asso
ciate a sequence i x < i 2 < ... satisfying conditions (2) and (3) in (i), and write M(v) = {ix, i 2,
For every x = (xn) e X N, let
oo oo
= ( £ wnten))llr, v(®) = (
n= 1 n= 1
by (1 ) we have Define
v(oo) < co(x) < 2v(x).
SC = {x e X N: v{x) < oo} = {x e X N: co{x) < oo};
then (SC, v) is a p-Banach space. After replacing v by a suitable equivalent p-norm ||-||, we will obtain the required universal space SCpqm = (SC, ||*||).
Let
W = {x e Ж : со (x) < 1}, V = {x e SC : v (x) ^ 1} ; of course,
l\ 1/p
- I F c f c F .
Complementably universal spaces 137
For every v e v > u, define a linear map j v: X->°I by
\x if n e M(v), j v(x) = (xn), where xn =
[0 otherwise.
Then
v(x) = oo(jv(x)) for all ж e l
by (2 ), and so j v is an “isometric” embedding of (X, v) into (№, cd). Write Wv = W n jv(X) = ] ; ( { ж е 1 : ф К 1 } | .
Define bn = bn(v) by
b\ — Щг, bn = barn for n > 2 , where an = an(v) and b = 1 + a[. ISTote that
(9) .
n = l
«
(ii) The formula
00
^ (® ) = Jv ( £ M *n)» ж = W e ^
71 = 1
well-defines a continuous projection from 3C onto j v(X).
Let x = (a?n) e l The series ^ bnx{ converges absolutely in (X, v):
П
00 00
£ « ( M i J = + &p £ aqnv(xin)
n ~ l n = 2
OO 00
< 2 У (® J < 4 V < -4 c , (a?, ) (by (3))
71 = 1
7 1 = 1 71 = 1
<40оо(я) (by (7)).
Hence
(10) (ü(Pv{<b) ) = v ( I >
71 = 1
Therefore P v is a continuous linear map from 3£ into j v(X)) that it is onto and idempotent follows from (9).
138 В. S o l t y s
Гог 0 < d < 1, let К й b© the p-convex hull of the set L d — dVи U {WV’ : v' e J f p, v' > w};
clearly, dV <= K d <= V. We are going to show that P v(Kd) c Wv for all v e Жр, v > и, if d is small enough.
F ix v e jVp1 v > u.
(iii) I f d ^ (8/?)-1/i>, then P v(dV) <=. Wv.
This follows from (10).
(iv) P V(WV') c= Wv for all v e / j,, v' > u.
We keep the notation an = an(v), in — ini'0) and К — bn{v) for the sequences associated with v; for we write: an = an(v'), in = in(v').
Let y e Wv>; thus y = j V'(oo), where vf (a)) < 1. Then PviV) *= P M = %(®)i
where z = (zn) with zn = œ for n e M(v)nM (v') and zn — 0 otherwise, and
« = P , ьп- n:ineM(v')
First consider the case tx ф г х. Since by (3) and (6) we also have win < anv < и < axv' < wti for n > 2, it follows that ix $ M (v).
Now
v(z) < co(z)< (J£w rn(®))1/f
* 71 = 2 oo
(by (3))
■ 71 = 2
< allrv'{æ) (by (5))
< « 1/r< ( 8f ) - 1 (by (8 )).
From (iii) it now follows that P v(y) = P v{z) g TFv.
Now let us assume that ix = i x. Let Jc be the smallest integer for which ik $M(v')\ Then
cq < c < 1 — bk = 1 — bark (b = 1 + ar) and
oo k—1
cor(Pv(y)) = cor{cjv(oc)) < ( 1 - K ) ^ w jn(a?) = ^
71 = 1 71 = 1
Complement ably universal spaces 139
where
00 00
00 oo
< (®) [ £ < - (1 + arK (ar + 1
X
a»)\ <ЪУ (3))n = k
= ^ И С | 4 - | ( 1 + ^ ) Ч ]
= 1 ^ ( ж ) [ з ~ ( н « т к о .
(by (5))
Therefore
fc-i fc-i
*>r(Pv(y)) < £ wrin(x) = wrln{x) < < 1 i.e., P„(ÿ) e Wv.
Let d = a1/a. In view of (8 ), (iii) and (iv), we have P v(K d) c: Wv, and hence K dn jv{X) — Wv and P v(Kd) a K d for all v e JTp, v > u. There
fore, if || -|| is the p-norm on 3C defined by
then for every v e J f p, v > u, the map j v is an isometric embedding of (X, v) into &рл>т — (&, ||*||) and P v is a projection of 3VPtQim onto j v(X) with l|PJj = 1 .
Finally, if v e J f p is arbitrary, we find c > 0 such that cv > u, and then
is the required isometric embedding of (X, v) into 9£р л ж Observe that since dV cz K d с V, the p-norms v and ||*|| are equivalent.
I t remains to verify that 3Cp<^m, or equivalently (Æ1, v), is isomorphic to the Banach space la. Since
it sufficies to know the following (rather standard) fact.
(v) There exist positive constants K , L (depending only on m — dim X) such that i f v e X rp (X )f ther.
for some norm uv on X such that {X, uv) and l™ are isometric.
Indeed, using this we may choose for each vn a norm un on X so that K u * < vn < L u i and (X, un) is isometric to Vf. Then the p-Banach space
||a>|| = in f {s > 0 : x e sllpX d}
00
№ y) = (©<*> *»))„
K u l < v < L u f
00
(W, / . ) = ( © (X, < )),
140 В. S o l t y s
is isomorphic to (ЯГ, v). On the other hand, (^, p) is isomorphic to the Banach space
(9, p'») = ( 3 (X, «„))» <ï=î>r) which in turn is isometric to lQ.
Bor the sake of completeness we will sketch a proof of (v). Let v A = {x: v{x) < 1}, 'B = convM, and let w be the Minkowski functional of B. Then
(11) + cz A cz В
which implies
(12) wp < v < {m J\-l)1~pwp .
To show (11), let x e (m + l f p~1)/pB. Then, by a well known result of Carathéodory ([5], Lemma following 3.25),
m +1
x — (m + t{xi3
i = l n
where x{ e A, f > 0 and У t{ = 1 . Hence
i=l
m '+1
v (x )^ (m + l ) p~1 Jj? 1,
i = l
and so x eA.
By Auerbach’s lemma ([4], p. 16) there exist norm one vectors et in (X, w) and e* in (X, w)*, i — 1 , .. ., m, such that e* (ef) — 1 if i = j and 0 if i Ф j. Define a norm uv on X by
✓ m
uv№) = ( £ \<>i (я)\а)11а•
ч i = l
Then, using Holder’s inequality, we have
(13) m~llauv фго ф mllquv,
where l f q + ljq ' = 1 .
Combining (12) and (13), we see that the norm uv is as required in (v), with К — m~plq and L — (m + lŸ~pmplq’.
A quick examination of the above argument shows that only slight changes in part (ii) are needed when p — 1 and q — 1 .
This concludes the proof of the theorem.
Problem. Given p e (0,1] and m e N, characterize those Banach spaces which are isomorphic to a p-Banach space complementably universal fo r
^ p3m *
Complementably universal spaces 141
I would like to express my gratitude to Professor L. Drewnowski for his help in preparing this paper.
References
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[2] W. B. Jo h n s o n and A. S zan k ow sk i, Complementably universal Banach spaces, Studia Math. 58 (1976), 91-97.
[3] N. J . К a lt on, Universal spaces and universal bases in metric linear spaces, ibidem 61 (1977), 161-191.
[4] J . L in d e n s tr a u s s and L. T z a f r ir i , Classical Banach spaces I , Springer-Verlag, Berlin-Heidelberg-New York 1977.
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