ROCZNIKI POLSKIEGO TOWARZYSTWA. MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXV (1985)
J. Cie m n o c z o l o w ski and W. Or l ic z (Poznan)
Variation and compactness
Abstract. Some relations are investigated between relative compactness of the sums £ rç. x, 1 and perfect convergence (subseries convergence) in either a Banach space or an F-space under some additional assumptions.
The condition 0 {X , У) is introduced as a generalization of the condition О which ap p e ars
in the theory of perfectly convergent series.
Applications of these notions to the study of some properties of vector valued functions of bounded weak variation are given.
00
1. Let (X , || U) denote an F-space. The series is called
l П
perfectly bounded in X if the set S of sums of the form £ xh n
i = 1
00
= 1, 2, . . . , rii = 0, 1, is bounded in (X , || ||). The series £ x t is perfectly l
convergent (subseries convergent) in (X , || ||) if every series of the form
00
]T tfi x i’ *1 i = is convergent in (X , || ||). We say that (X , || ||) satisfies the l
00
condition 0 if every perfectly bounded series xf is also perfectly convergent.
l
We are going to give some generalization of the condition 0 . Let (X , || ||), (Y, || II*) be F -spaces, X c Y The space X is said to satisfy the condition
00
0 ( X , Y) if every perfectly bounded series ]T xh x, e X in X is perfectly l
convergent with respect to the norm || ||*. In connection with the condition 0 we give in Section 1 of this article a generalization of Gelfand’s criterion
00
for the perfectly bounded series £ x t to be perfectly convergent. In Section 2 i
we give an application of the conditions О and 0 (AT, Y) in the theory of vector valued functions of bounded weak variation.
1.1. Lemma 1. L et (X , || ||) be either an F -space with a basis or a B-space.
202 J. C i e m n o c z o l o w s k i and W. O rl i c z
I f the set o f elements х,- e X is relatively com pact in X and the series Y xf is 1 perfectly bounded, then xt —► 0.
P r o o f. Let us assume that the elements щ constitute a basis in {X , || ||) 00
so that x = Yj aj{x )u j, x e X , where aj(x) are linear functionals continuous in
i m n
{X , || U). Let Am(x) = Y aj(x)Uj for m = 1, 2, ... Since the set yK — Y rç,*,-,
J=1 i=l
n = 1, 2, . . . , r/i = 0, 1 is bounded in (X , || ||) so the sequence aj(y n) П
— Y */« aj (*i) is bounded for j — 1, 2, ... and arbitrarily chosen rj{ = 0, 1. In i— 1
consequence
(A) X \aj ( x i)\ < 00 for j = 1, 2, ...
i = 1
Thus а7(х ,)-> 0 for 7 = 1 , 2 , . . . and so Aw(x,)-^ 0 when i-> oo. One can extract from (x,) a partial sequence (xk.) such that for some x e X , ||xk. — x||
0. Thus Am(xk.) -> Am(x). On the other hand Am(xk.) -> 0 so Am(x) = 0 fo.r m = 1, 2 . .. and since Am(x )-+ x so x = 0 and consequently xk. -> 0. Because the above reasoning can be carried out for arbitrary partial sequence of (x,) so xf -> 0. In the case where (X , || ||) is a Banach space, the perfect
00 00
boundedness of the series £ x t implies £ |£(x,-)| < oo for arbitrary functional
i i
from the dual space of (X , || ||). For some partial sequence (xk.) there exists an x such that xk. x so £ (xk.) Ç (x). Since, on the other hand £ (xk.) -> 0 so
^ (x) = 0, x = 0 or xk. -> 0. Applying this reasoning to arbitrary partial sequence of the sequence (x,) we conclude x,- -> 0.
1.2. Theorem 1. I f (X , || ||) is an F -space with a basis, then the series 00
£ x,- is perfectly convergent i f and only i f the set S is relatively compact in 1
(X , || U) (i.e., the closure o f S is com pact in (X , || ||)).
P r o o f. To show the necessity let us choose an arbitrary sequence yk
nk oo
= Yj Vik)xi = Y x i> where rj\k) = 0 for i > nk. Using the diagonal process we can extract a sequence of indices (k v) such that щ v -► rj\0) for every i as kv ~* oo. Any perfectly convergent series in an arbitrary F-space has the following property: for every e > 0 there exists an index N such that
III 4 for p ^ N and arbitrary rji = 0, 1. The sequence Tkv
i=P00 00
= Y */i*v) x,- converges to Y ^!0)х, - Indeed, choose arbitrarily e > 0, then
i i
N— 1
E rçjkv)x ,-» £ rçj^x,- and also ||£ < е for all /cv. In consequence N— 1
I1 (*v>
we get
N- 1
IE »й0,х,|| ^ Il I (4,“,,-ill0,)*(|| + ||I 0),(V->T)*,|| « 36
1 1 1 N
for sufficiently large k v.
Sufficiency follows from Lemma 1 if we consider the remark that any sequence of sums E щ х ь щ < m ,< ni+ 15 is perfectly bounded and relatively
ni
compact.
00 13. Theorem 2. L et (X , || ||) be a Banach space. The series E xf is
l perfectly converqent in (X , II ||) i f and only i f the set S is relatively com pact in (X , || H) (cf. [5] for a different proof).
P ro o f. Necessity follows from Theorem 1. Sufficiency is implied by an application of Lemma 1 like for an F -space.
Let (X , || II) be a Banach space, E its dual. The set E 0 <= E is called fundam ental or norming if the set {||£||} is bounded for Ç e E 0 and, with some c > 0, ||x|| c ^ sup |£(x)| for x e X . The set S j cz E is relatively *-w- sequentially compact in E if from every sequence Çne E t one can extract a partial sequence (£„.) *-w-convergent, that is to say, (x )-> £ (x) for x e X . The set X 0 a X is relatively weakly sequentially compact with respect to E 2 if from every sequence x „ e X 0 one can extract a sequence (x„.), S 2-weakly convergent to some element x, that is to say, £(x„.)-> £(x) for Ç e S 2-
00 Theorem 3. L et (X , || ||) be a Banach space. Assume that the series E xf
l has the following properties: the series is perfectly bounded; f o r some fundam ental set o f functionals S 0 which is relatively *-w-sequentially compact,
the set S is relatively w eakly sequentially com pact with respect to E 0. Then the 00
series E xt is perfectly convergent.
l 00 00
P r o o f. Perfect boundedness of E xi implies E |£(xf)| < oo for Ç e E 0.
l i
00 "fc
For arbitrary E *7, *,, rji = 0, 1, there exists a partial sequence E rç.x,- such
1 i
nk
that E rii Ç (x,) -+ £ (x^) for Ç e E 0 (x^ independent of £). Therefore
l 00
(a) E It £ (*,) = £ (*„) for £ g S0, rji = 0, 1.
204 J. C i e m n o c z o l o w s k i and W. O r l ic z
By virtue of Theorem 1 in [9 ] the perfect convergence of £ xf follows.
i
R e m a r k . If in the theorem above we set £ 0 = 5 , ||£|| ^ 1 then the perfect boundedness is a consequence of relative weak sequential compactness of S with respect to 3 0. The elements xn of (a) belong to the separable linear space X 0 spanned over S and so 3 is relatively *-w-sequentially compact in X 0 and in this case the above theorem may be applied, (a) means here the assumption of the Orlicz-Pettis theorem.
1.4. Let (X , || К) be a Banach space, (Y, || ||*) an F -space, X <= Y. The unit ball X s = \ x eX : ||x|| ^ 1} endowed with the metric d(x , y) = ||x —y||*
provided it makes X s a complete metric space, is called Saks space and denoted (X s, || II, || II*) (see [8]).
Theorem 4. L et (X, || ||) be a Banach space, (Y, || ||*) an F -space having a Schauder basis or a Banach space, X <= Y. I f ( X s, || ||, || ||*) is a relatively com pact Saks space, then X satisfies the condition 0 (X ,Y ).
. 00
Pr o o f . Indeed, if the series £ x, is perfectly bounded in (X , || ||), then i
the set S is bounded and for some к > 0, kS cz X s. It suffices to apply Theorems 1 and 2.
1.5. We are going to give two applications of Theorem 4.
(i) Let co(u): <f0, oo) -> <0, oo) denote a continuous strictly increasing function, <y(0) = 0. Denote by X = H w the vector space of real functions in (a , b } taking 0 for t — a and satisfying the condition
(*) |x(r1) - x ( r 2)| < /ccu(|r1- t 2|) for tu t2 e {a , by.
Let us define a standard norm in X, ||x||w =inf/c (k from inequality (*)).
(Нш, || ID is a Banach space. Note that if (o{u)/u -» 0, и -» 0, then H w is reduced to the function x(t) = 0 for t e ( a , h y . Let Y = ( H W1, || Ц^, ), where aq satisfies the conditions from the definition of со and the additional one:
co(m)/coj (m) —► 0 where и -> 0. It is easy to see that X c= Y. Let us form the Saks space (X s, || ||ю, || ||w ). Its completeness follows directly from the observation that ||x„ — х 0||ю -+ 0 implies x„(t) — x 0 (t) -> 0 uniformly in (a , by, and liminf HXj. - xX j ^ ||хи-Хо||Ш1, liminf||x„L ^ ll^olL- We shall prove
m oo n -* oo
that (X s, || l^, || ||Ю1) is compact. Let x ^ X , = {x gHm: ||x||w < 1]. From the Arzelà theorem there follows the existence of a partial sequence (x„.) such that x n.(t) -> x 0 (f), x 0 e l s, uniformly in ( a , by. Put y((f) = x„.(t) — x 0 (t), then
\ y i(ti)-y i(t2)l < 2û)(|t i - t 2|) = 2 — ^ — T ^ W i d f i - ^ l ) .
Taking m0 > 0 sufficiently small we have, for \tl — t2\ ^ u 0,
LM'i) - ) ;.(*2)I < «М |п ~h\\ i = 1 , 2 , . . . For arbitrary tl5 t2 for which
\tl - t 2\ > u 0 we have LM*i) - . M ï2)I ^ £<*h («о) < (l*i - t 2\) for i ^ i0 . In consequence ||x„. — х 0||ш -* 0, while x 0 e X s.
Putting in particular w(u) = ия, o)x (m) = up and denoting in this case
= H f н шх = H p, Il \\a = Il ||e, II ||л,1 = II 11^ we can see that the Saks spaces (A^, Il IL II II*») are compact for arbitrary 0 < < a ^ 1.
In connection with this theorem, let us notice that fail the condition О for 0 < a ^ 1. For a < 1 this follows from a theorem of Ciesielski [4 ] about the linear isomorphism between Ha and /*; for a = 1 this is a consequence of linear isometry of H l and L°°<a, b>. However, Theorem 4 insures that, for 0 < f < a < 1, 0 ( H a, H p) holds.
(ii) Let a ^ 1. The real function x in <(a, by is called function o f bounded ct-variation if
ua(x) = sup £ |x(f() - x ( f i_ 1)|e < oo, /= l
where the supremum is taken over all partitions я: a = t0 < t1 < ... < t„
= b.
The set Vя of functions of bounded a-th variation, taking 0 for t = a, is a Banach space with the usual operations and with the norm ЦхЦ,,^
= (ra(x))1/ot. It is known that V1 satisfies the condition О whereas it is not known whether this condition still holds for a > 1 [1], [7].
Let a > 1, œ the same as in 1.5 (i), X = V f Y = Vfi, where a < jS.
For X take the B-norm ||x|| = sup(||x||w, ||x||„,e). Then X <= Y, (X s, || ||, || \\Vtfi) is a compact Saks space.
For the proof let us take a sequence x „ e X s — { x e X : ||x|| ^ 1]. Since HxJL ^ 1 so there exists a partial sequence (xff.) such that x„.(t) -* x 0(t) uniformly in (a , by. Since liminf ||x„.|| ^ ||x0|| so ||x0|| < 1. Put y ft )
i * oo
= x n.{t) — x 0 (t) and choose partitions я£: a = t\ < t l2 < ... < tlMi) = b in such a way that
n(i)
( U ) i f l l v J L / < I Ы ^ - y t i f j - i Ÿ for / = 1 , 2 , . . .
j = 1
The functions yt are equicontinuous in (a , by; hence taking u0 > 0 sufficiently small, we have |.у£(ф —)>,•(/}-i)l ^ e 1/p~ct for such intervals in the partition я,, where \t) — r}_ 41 < u 0. Let the set of these intervals of щ be denoted by А (я,) and the remaining ones by В (я,). Let kt denote the number of intervals in В (я,). O f course, <{ b — a)/u0. In view of uniform convergence of y£ to 0, summing over intervals of В (я,), we get
(1.2) X \уА Ъ ) -уАЪ-1)\р < £ f o r i ^ / o - В(л$
2 — Prace Matematyczne 25.2
206 J. C i e m n o c z o l o w s k i and W. O rl i c z
However, for i = 1, 2, . . . , we have
(1.3) I £ 1л(»})-Л(«}-1Г«Н’, ( л К 8(1 + 2“).
A(nt) A(7ij)
Inequalities (1.1H 1.3) yield \\yt\\v,f i (28(1 + Т ))'Ш for i > i0. So ЦЛ ||„.Д - 0. (It is apparent that compactness of X . implies completeness of the Saks space
(xs, Il H, Il IL»
00
1.6. Theorem 5. L et xt e C fo r i = 1, 2, ... and f o r the series x, let the 1
set S be relatively com pact in C with respect to convergence in measure. Then 00
£ xf is perfectly convergent in C.
i
P r o o f . Since the set S is relatively compact in C with respect to convergence in measure so it is relatively compact in L° and thereby, bounded in L°. From this there follows the perfect boundedness of the series
00
£ xf which implies, by virtue of Theorem 1 of [6] its perfect convergence in
i 00 lk
L°. For a given series £ some partial sequence £ rç,xf converges in
1 i
00
measure to xn e C, which means £ xf = xn with respect to convergence in i
measure for arbitrary rjt = 0, 1. By virtue of Theorem 7 of [9 ] the perfect
00
convergence of £ xf in C follows.
i
2. In this section we shall be interested in the condition 0 ( X ,Y ) for some spaces of sequences. Let 1° denote the vector space of sequences with real terms x = (t,). Let q> be a ^-function, i.e., tp: <0, oo)-+ <0, oo), <p(0) = 0,
<p{u) > 0 for u > 0, <p continuous and nondecreasing, <p(u)-> oo for и -> oo.
00
In 1° a modular ^ ( x ) = £ <p(|t,-|) defined. Write l*v = {xe/°: £v(Ax) < oo i
for some A < 0}, IJ* = {xe/°: ^(A x) < oo for all A > 0} — the space of finite elements. I*9 is an F-space with respect to the generated norm ||x|L
= i o f [ e > 0 : Q(p{ x / s ) ^ e } , and tf* its complete subspace. | | x J L 0 is equivalent to (?*(A x J-» 0 for each A > 0. Recall also that q> satisfies the condition A2 for small u if for some к, u0 > 0, 0 < и < u0, <p(2u) ^ k(p(u).
2.1. Lemma 2. I f IJ* satisfies the condition O, then tp satisfies A2 fo r small u.
P r o o f . Suppose A2 fails. Then there exists a numerical sequence u„-> 0 such that (p(2un) ^ 2n(p(un), q>(un) ^ 1/2", (p(2u„) ^ 1 for n = 1, 2, ...
Determine integers k„ in such a way that l/2"+1 ^ k n(p{un) ^ 1/2" and define
sequences x x = {ut , ul 9 . . . , uu 0, 0 , x„ = (0, 0, 0, u„, un, . .. , u„,
where the i-th term of x„ is equal to u„ for + ... + /c„_ ! < i ^ k x + ... + k„, and Xj has iq at i, 1 < i < f c j . Let us choose a sequence Аг -» 0. We have x„eI*<p,
* i + ••• +Г1кХк)) 00
/сп<р(Ягии) < e for i ^ i(£), к = 1, 2, ...,r ji = 0, 1,
tl~ 1
0 0
because £ /c„<p(u„) ^ 1. Suppose HxJ^-oO; then ^ (2 x „ )-> 0 but p„(2x„)
n = 1
^ k n(p(2un) ^ j — a contradiction. So the sequence (x„) is not convergent in l**
00
which means that the series x„ is not perfectly convergent, being at the same l
time perfectly bounded.
2.2. Lemma 3. T he space l*v has a basis consisting o f unit vectors e{ (A 2 is not assumed).
00 00
P r o o f . If x e IJ* then, for arbitrary A > 0, £ <p(Ati) < oo, £ <р(Лг.) -> О,
1 i = k + 1
that is, for the sequence xk = (0, 0, . . . , tk+1, tk+2, • ••), 11**11* 0 whenever к
к oo
oo, ||sk-x||„ -> 0, where sk = £ f e j , and x = £ */*/•
l l
2.3. Theorem 6. Let П
(*) e * £ */***) fo r n = 1 ,2 , . . . , rjk = 0, 1.
1
Then
П
(**) Qv ( i ( Z *7k xk ~ *>,)) ^ 0 f ° r arbitrary rjk = 0, 1,
where Q ^ d x J ^ K .
P r o o f . Let us introduce some notations: Xe ~ the characteristic function on the set N of natural numbers; if e = <p, q } then we write Xpq instead of xe \ X(p,ao) wiH have parallel meaning. We assert that for each £ > 0 there exists a p such that
(2-1) M i * * 2 < P,ao)) < £ for к = 1, 2, ...
If 2.1 does not hold, then there exists a partial sequence of (xk) denoted (y,) and a sequence of intervals of natural numbers ph q h Pi < qi < pi+1, for l = 1^ 2 , . . . and £0 > 0 such that
(2.2) QP(iyiX<piqi> ) > eo> 1 = 1 , 2 , . . .
208 J. C i e m n o c z o l o w s k i and W. O r l i c z
Let y t = (?•)•(*) implies
П
(2.3) (p ( £ rjt ф < K for arbitrary rjt = 0, 1.
/= î
Let (p(u) > K for и > u0. From (2.3) we get
00
(2.3') £ | r | K 2 u„ f o r i = 1 , 2 , . . . /= 1
Let y„ = rii y 1+ ri2 y 2 + ... +ri„yn, y„ = (t?). Define sets e\ = { i e <p,, q t) : ItfI < ilrill, e'i = </>/, 3<>\4. Whenever / -> oo then ph q t -> oo so (y„*P/4/) -► 0, and since
O ', X p ,,,) ^ ^ (У * X e ” ) ^ Q * ( i У i X e ” )
so, for sufficiently large /, ^ i«o» that is
( i Z c;) = 6* ( i Уi X(pfil>) ~ Qv ( i Уi * e”) > î «о.
and there is
(2.4) |*?| when /ec,'.
Let a,' be such a subset of e\ on which |rj| > 0 and let liminf|r{J = dt. Clearly
ieaj
(2.5) Q9 (iyiXa'} > i * o ,
and furthermore (2.4) holds for i g a\. Having fixed some / for which (2.4), (2.5) i e a[
is valid we can always find an lx in such a way that
(2.6) It!1! rjdt for iea'h
where rj is given in advance. The possibility of choosing such an lt is implied directly by (2.3'). Using (he remarks above we are in a position to define a partial sequence (y,k), y lk =(t\k) and a sequence of sets (uk) satisfying the conditions:
1° e 9 (iy ik Xak) > U o >
2° I X t\J\ < for i e a k,
7 = 1
oo
3° > X for i e a k, j = k + l
4° ak- n aw. = 0 when k' # k".
From l°-3° we obtain
еЛ(£ й,)ч)=Е И Х »?|)> I
j = 1 ieak j - 1 »«»k
Hence by (*) and 4°, denoting br = [j ak, we have К ^ M ( £ у».)*ь ) ^ |e0 r
k=l j=l
so taking r > 4Х/Зг0 we are led to a contradiction.
00
Because of (2.1) there is ]T <p(i|f?|)^e for sufficiently large p,
i — P
00
к = 1 , 2 , . . . , and (*) yields ф ( ]T |tf|) ^ <p (2u0) for i = 1, 2, ... Thus for к ^ k 0
k = 1
there is M i * * ) < 2e that is M i * * ) - * 0. Let us consider an arbitrary series
00 n
X rçkxk, s„ = X Чкхк- The Cauchy modular condition
1 l
(2.7) qv ( i (sp - s,)) -> 0 as p, q -+ go
is satisfied. Otherwise there would exist, for some £0 > 0, an increasing sequence of indices ph qh p( < qt < pi +1 for / = 1 , 2 , . . . such that 0 * ( i( s«i- s Pi)) ^ £o- However, for the sequence Z i= s q. - s Pi, by (*) we
П
have M X rjiZi) ^ К for n = 1 , 2 , . . . and arbitrary — 0, 1. Using the l
previously proved lemma for zf replacing xh we would have pv (iz,-)-> 0 and we are lèd to a contradiction. From (2.7) there follows the exist-
n
ence of a sequence xn — (tf) such that, for each i, ]T rç* t* -* t f and i
liminf M i ( sp — s9)) ^ M i ( sP — дс,)). From this last inequality and (2.7)
q -*oo
we get M i ( sp~ M ) < £ for p sufficiently large and thus (**) holds. Since
M i* , ) ^ M i (sp “ *»/)) + M i sp) < e + K for p sufficiently large, so
2.4. Theorem 7. L et q>, ф be (p-functions satisfying the condition: fo r each X > 0 there exist cx > 0, u(X) such that
(*) ф(Хи) ^ c x (p(u) fo r 0 ^ и ^ u(X).
Then l*9 c= If* and О (l*9, 1*ф) holds.
<X
P r o o f . The inclusion l*9 о If* is obvious. Let the series X x, be l perfectly bounded in l*9. Then, for some A0, there is
(2.8) M M X */,■*,•)) < 1 for n = 1, 2, . . . , rji = 0, 1.
1
n n
Hence (p(X0 X rç,fk) < 1, and for some u0, X0 X! Kiel ^ uo for к = 1, 2, . .. ,
i=l i= 1
210 J. C i e m n o c z o l o w s k i and W. O r l ic z
n = 1 , 2 , . . . However, (*) implies that for some constant cx > 0 ф(Хи) ^ cx (p(u) for 0 ^ и ^ u0, so (2.8) yields
n n
(2.9) (ÀÀ0£ rii Xi) < c x Q", (A0 £ % Xi) < CA.
l l
From (2.9) and Theorem 4 we obtain for some x£
П
(2.10) Q+ (i U.Q ( £ r\i xt - xj)) -♦ 0 for each A, i = 1
but it is clear that x* = x^ independently of A , and in this way from (2.10) we
n
get |E m x i - x n\|*->0.
i
R e m a r k 1. In the above theorem we did not assume (p, ф satisfied A2 for small u.
2. If (p does satisfy A2 for small u, then (*) holds when we set (p(u) = ф(и). In this case, l*4* = 1*ф and for the space l*v the condition О is fulfilled.
3. Theorems analogous to Theorems 6 and 7 for function spaces I f 9, Ь*ф have been given in [3].
3. Let (X , || If) be an F-space, x: <a, b } -> X. The function x ( •) is called function o f bounded weak variation whenever the set of sums
Л
S(n, n, rj) = X Vi (x(h) x(f j _ i)), i= 1
where n : a = t0 < t l < ... < t„ = b is an arbitrary partition of (a , by, rji
= 0, 1, n = 1 , 2 , . . . is bounded in (X , || ||).
Denote VW(X) the set of functions x(t): ( a , b } - + X , which are of bounded weak variation and x(a) = 0. With the standard operations on functions it is a vector space. If AT is a Banach space (s-normed complete space), then X 6 k w( l ) if and only if vw(x) = sup||5(n, n, rj)\\ < oo, where supremum is taken over all n, n, rj. In this case t>w(x) is called weak variation of x and VW(X ) is a Banach space (s-normed complete space) if we set ЦхЦ,,
= uw(x).
Let (Y, || U*) be an F -space, X c Y We are interested in the following properties of functions from VW(X ):
A. Let X6 VW(X). At each point of <a, by there exist one-sided limits of x (right-sided at t = a and left-sided at t = b), with respect to || ||*
convergence ;
B. Let x e V w(X). The function x is continuous in {a , by except for a countable set and with respect to the norm || ||*.
Theorem 8. L et (X , || ||), (У, || ||*) be F-spaces, X a Y. I f x e VW{X) and has property A, then it has property B .
P r o o f . Denote
cl>(0 = limsup {||x(Y) — x(f")ll*: t', t " e ( t — à, f + <$>}.
6 -*o
We assert that the set Ak = {r: co(t) ^ 1 /к ], к = 1 , 2 , . . . , is finite. If the set Ak was infinite, then there would exist some t0 and a sequence of different tf ->• t0, h E ^ k; we may assume, for instance, t{ > t0. Let us choose a sequence of disjoint intervals <t, — <5f, ^ + <5,-> and points fj, t'/e — h + &,> in such a way that ||x(t-) — x(f,")||* ^ 1 /2А: for i = 1, 2, ... However, t\->t0, t'{ -* t0 so,
GO
by property A, ||x(fj) — x(t-')||* -* 0 and we get a contradiction. The set у Ak k= 1
00
is at m ost countable and for /е <д, b}\\J Ak the function x is continuous l
with respect to || ||*.
Theorem 9. L et ( X , || ||) be an F-space, (Y, || ||*) either a Banach or an F -space with a basis and let || || be not w eaker than || ||* in X, X c Y
A function x e VW(X) has property A i f and only i f the set o f its values is relatively compact in (У , || ||*).
Pr o of . Necessity is obvious: assuming A from every sequence (t„) a partial sequence (r„.), tn /* t0 (or tn. \ t0) can be extracted such that x{t„.) -* x 0 with respect to || ||*.
To prove the sufficiency suppose that for a < t0 < b the left-sided limit of x(f) in (Y, || U*) does not exist. Then there exists a sequence t( A t0 and a sequence of disjoint intervals <tf tjk) such that ||x(tifc) — x (tjk)\\* > e0 for к
00
= 1 , 2 , . . . Since x e VW(X ) so the series ^ (x(rijk) — x(t^)) is perfectly bounded l
in (X , || II) and thereby in (Y, || ||*), using Lemma 1 for (Y, || ||*) we get
||x(fi ) —x (ïj-)||* -* 0 and a contradiction.
The following simple theorem has already been proved in [3] but as it concerns property A and for the reader’s convenience we formulate it here again with the proof.
Theorem 10. L et (X , || ||), (Y , || ||*) be som e F-spaces X c Y
Each function x e VW{X) has property A i f and only if the condition 0 ( X , Y) holds.
CO
P r o o f. If 0 ( X , Y) fails, then there exists a series £ x, perfectly bounded l
in (X , || II) and such that ||xf||* ^ e 0 for f = 1, 2, ... Let (w,) be the sequence of rational numbers in (a , b ) arbitrarily arranged. Define a function x setting x(f) = xf for t = Wi, x(t) = 0 elsewhere in (a , by. It is easy to notice that x e F w(x) but neither left-sided nor right-sided limits exist at any point of (a , by with respect to || ||* convergence.
212 J. C i e m n o c z o l o w s k i and W. O rl i c z
Sufficiency is obtained by reasoning analogous to the proof of the preceding theorem and application of 0 ( X ,Y ) instead of Lemma 1.
Theorem 11. L et ( X , || ||) be a Banach space, E 0 — a countable set in the dual space E. I f x e VW(X) then, except fo r a countable set, x is *-w- continuous with respect to E0.
Pr oof . Let (£„) = £(,. We introduce a B 0-pseudonorm in X 11*11* £ 1 |g,(*)l
1 2" 1 + l ^ w r
Let co(t), Ak, etc., have the same meaning as in the proof of Theorem 8.
We are reasoning analogously to its arguments that if Ak is infinite then
ll*W )-*(tf)ll* > 1/2к for some t0, t\, t'- -> t0 and disjoint (t[, t”} . However, 00
since the perfect boundedness of the series ]T (x(fj) —x(f-')) in (X , || ||) implies i
~ * ( O) 0 for « = 1, 2, . . . , that is, ||x(tj) — x(f-')||* -> 0, we have a 00
contradiction. For t e ( a , b}\\J Ak, x is *-w-continuous with respect to Н0 . i
Theorem 12. L et (X , || ||), (У, || ||*) be Banach spaces, X cz Y, T0 — a countable fundam ental set in the dual space o f (Y, || ||*). L et the norm || || be not w eaker than || ||*. I f x e V w(X) and the set o f its values is relatively *-w- com pact with respect to T0 in (Y , || ||*), then x has property A fo r the *-w- convergence in (Y , || ||*) with respect to T0.
P r o o f . Let (t„) = T0. Define a norm ll*li° = X 1 M * ) l
2" 1 + |ти(х)Г
Applying the reasoning analogous to the proof of Theorem 9 we find out that |jx(tp) — ^'(tq)||° —* 0 as tp, tq -> t. However, the assumption of compactness of the set {x(r)| in (Y, || ||*) for the *-w-convergence with respect to T0 means compactness with respect to the norm || ||°. This and the Cauchy condition yield ||x(fp) —x 0| | ° 0. Note that in Lemma 1 one can take a B 0 space instead of Banach space and restrict oneself to the fundamental set of functionals.
Theorem 13. Let X — C (c , d } , x e VW(X) so x,( •) = x (t, u) is continuous with respect to и fo r each t. x = x (f, u) has the follow ing properties:
(*) fo r each и the function x(t, u) is continuous except fo r a countable
set (depending on u), 1
(**) let y,+ (u), >•' (u) denote, respectively, the right-sided and left-sided limit o f x (t, u) at t.
T here exists a countable set D such that, fo r t0e (a , b}\D, x (t0, u)
= V’,q(w) = yfo(w) except f o r a set o f m easure 0 (depending on t0), i.e., x ( •, u) are continuous f o r almost all и on <a , b}\D.
Pr o of . From finiteness of vw(x) there follows \ x(t,u )\ ^ K so y * (u) is a measurable and bounded function. Let us choose the set E 0 of
d
functionals over C ( c , d } of the form \ h(u)y(u)du, y e C ( c , d } , S 0
C
countable and norming with h(u) integrable. According to Theorem 11,
d d
except for a countable set, lim | x(t, u)h(u)du = j x (t0, u)h(u)du. But
f ^ f o 'c
d d
there is also lim J x (t, u)h(u)du = j yfQ(u)h(u)du. Hence x (t0, u) — y^0 (u)
c
for almost all u. Reasoning analogously for t -» tô we get x (t0, и) = yt~ (и) except for a countable set and for almost all u.
n
3.1. Theorem 14. Let x: (a , b>-> C, the set o f sums X rii (x(ti) — x (ti- l ))
i
f o r arbitrary n, щ = 0, 1 and any n be relatively com pact in C with respect to convergence in measure. Then x e VW(C) and x has the property A with respect to the norm in C.
Pr o of . For some (a , /?> c (a , b ) denote by vw{x; a, f ) the weak variation of x (here X = C). We are going to show that vw(x; a, b) < oo.
Suppose vw(x ; a, b) = oo and note the following lemma: if for some t0, a < t 0 < b , vw(x; a, t0) < o o , vw(x; t0, b) < oo, then vw(x; a, b) < oo (cf. [2]).
The set is bounded in C. Otherwise there would exist a sequence of non-overlapping intervals <a,, /?,) such that the series
00
(3.1) £ (*(&•)-*(<*.•))
i
would be divergent in C. Using the assumption on x and Theorem 5, we get a contradiction. Using the lemma given above and boundedness of the function x we can (cf. [2]), define a sequence of intervals (cth /?,) for which series (3.1) is divergent in C and so, as before, Theorem 5 leads to a contradiction. Property A follows by a simple argument like in Theorem 9 and by repeated application of Theorem 5.
References
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[2] J. C ie m n o cz o lo w sk i, W. O rlicz , Inclusion theorems fo r classes o f functions o f generalized bounded variations, Comment. Math. 24 (1984), 181-194.
[3] —, —, On Some Classes o f Vector Valued Functions o f Bounded Weak Variation, Bull. Acad.
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[5] I. G e lfa n d , Abstrakte Funktionen und lineare Operatoren, Mat. Sb. 4 (1938), 235-286.
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Sci. Math. Astronom. Phys. 16 (1968), 801-808.
214 J. C i e m n o c z o l o w s k i and W. O r l i c z
[7] W. O rlicz , Une généralisation d’un théorème de MM. S. Banach et S. Mazur, Ann. Soc.
Polon. Math. 19 (1947), 62-65.
[8] —, Linear operations in Saks spaces I, Studia Math. 11 (1949), 237-272.
[9] —, On perfectly convergent series in some junction spaces (in Polish), Prace Mat. (Comment.
Math.), 1 (1955), 393-414.
INSTYTUT MATEMATYCZNY PAN
INSTITUTE OF MATHEMATICS, POLISH ACADEMY OF SCIENCES, MIEL2YNSKIEGO 27/29, 61-725 POZNAN
