Some Results On Matrix Spaces. II

Some Results On Matrix Spaces. II

MATHEMATICS SOME RESULTS ON MATRIX SPACES. II BY H. H. A. EL MAKAREM (Communicated .by Prof. J. F. Koxs:MA. at the meeting of April 28, 1956) (VIII...
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MATHEMATICS

SOME RESULTS ON MATRIX SPACES. II BY

H. H. A. EL MAKAREM (Communicated .by Prof. J. F. Koxs:MA. at the meeting of April 28, 1956)

(VIII) When IX<;U00 and IX*=O'v the sequence A 1"'1(m=l,2, ... ) is p-cgt in u ~IX if, and only if, (i) a number p exists such that a~1 =0 for every m and n, and for k>p, (ii) {.A 1"'1} is c-cgt, (iii) la~1 1 ..;;;Mk for every m and n, and (iv) the columns of the A 1"'1 are in IX. Proof. The conditions are sufficient. By (i), (iv) and (I), the A 1"'1 are in <1 ~IX. Let x be a point in u, and let y 1"'>=Ax; i.e., y~m> =

I

fl

k=l

a:> xk.

Let c -lim A 1"'1 = A """ (ank) ; then lim y~"' 1 = m-+-oo

• { y (m)} Is • I.e., c-cgt .

By (iii), IY~"' 1 j.;;;;;

fl

I

k-1

Ia:> x"l,;;;;

p

I k=

1

ank xk; (l)

p

I

k=l

lxki·Mk for every m and n.

(2)

By .(1), and (2), {y1"'1} is 0'00 -cgt 37 ). But {y1"'1} is in IX, and 1X*=<11 =<1~; therefore {y} is IX-cgt, and hence {A} is p-cgt in <1 ~IX. The conditions·are necessary. Since the A are in <1 ~IX, their columns are in IX 38 ). The fundamental unit vectors e< 1>, e<2 >, ... , e, ... are in u. Let z1"'1 =Ae; then z~"'1 =a:> and {z1"'>} is IX-cgt. But IX<;0'00 and IX*=0'1 =~; therefore {z1"'1} is 0'00 -cgt. Thus ja:>I..;;;Mk for every m and n. Also Jim a!Jl:1 exists for every n and k; i.e., {A 1"'1} is c-cgt. m-oo

Since A is in <1 ~ l,., by (I). Let u be a point in d, and let y =A 1"'1u; then {y} is IX- cgt. Therefore IY~"'1 I = 0 for every m and n and for k>p. Cor. The sequence Af"'1(m= l, 2, ... ) is p-cgt in <11 ~ cp if, and only if, (i) a number Po exists such that a~';:1 = 0 for every m and k and for n > p0 , (ii) {A 1"'1} is c-cgt, and (iii) Ia~> I .;;;;M,. for every m and k. .... 37) 38 )

CooKE, (1), p. 310, (10.6, IV). COOKE, (2), p. 299, (6.4, I).

500 Proof. By theorem (VIII), {A'} is p-cgt in a-+ a00 if, a.nd only if, (i) a'J:>=O for every m and k, and for n>p0 , (ii) {A'} is c-cgt, and {iii) la'J:>i =a~~>, a and a 1 are perfect, a*=cf>, ai=a00 , cf>*=a, and a~=ap Therefore the given conditions are sufficient and necessary for {A } to be p-cgt in a 1 -+ cf> 39). (IX) The sequence A(m= I, 2, ... ) is p-cgt in a-+ a 1 if, and only if, (i) a number Po exists such that a':!!!/=Ofor every m and n, and for k>p0 , (ii) {A} is c-cgt, and (iii) to an arbitrary E>O corresponds a number

fk such that

00

,L

n-tk+l

Ia~~> I< E

for every m.

Proof. The conditions are sufficient. By (iii), the columns of the • . (l) A are in a 1 • • By (1), (i), and theorem (I), the A are in a-+ a1 . Let x be a point in a, and let y=Ax, i.e.,

== (ank) ;

Let c -lim A = A • I.e.,

{

L

n-tk+l

la~>i

40 ),

y~m>

m-+oo

• y (m)} 1s c-cg t .

By (iii), when xk'7"'0 00

then lim

Po

L a,.k xk;

=

k~l

(2)

to E> 0 corresponds a number

f,.

such that

< Ej.(p0 ·lxkiJ for every m.

Let N>f,. for k..;;;p0 ; then

(3)

By (2) and (3), {y} is a 1 -cgt 41 ); and hence ~A} is p-cgt in a-+a1 • The conditions are necessary. The fundamental unit vectors e, ••• , e,, ... are in a. Let z =A e =a~~> and {z} isa1 - cgt. Therefore to an arbitrary for every

m.; Also I

E>

lim

,._,.co

0 corresponds a number

a~>

exists for every

n such that L

n and

co

n-tk+l

Ia~~)

I< E

k; i.e. {A} is c-cgt.

Since {A }. is p- cgt in a -+ a 1 , and in sequence spaces projective convergence implies projective boundedness, therefore {A } is p- bd in a-+ a1 . Following the lines of theorem (IV), we see that a number p exists such that a:>= 0 for every m and n, and for k > p. 39 )

COOKE,

40 )

When

41 )

(2), P· 326, (6.6, I).

x,. =

0, hence

lxkl·

00

~

n-N+l

CooKE, (1), p. 310, (10.6, VI).

la:>l

=

0.

501

Cor. The sequence A(m = 1, 2, ... )is p-cgt in a00 --+ 4> if, and only if, (i) a number p exists such that a~~'=O for every m and k; and for n>p, (ii) {AO 00

I la::Z'I < E

corresponds a number h.. such that

for every m.

k~h,.+1

Proof. By theorem (IX), {A'p, (ii) {A'tm>} is c-cgt, and (iii) for every fixed n and to an arbitrary E> 0 corresponds a number 00

I ja'l::!'l < E

hn such that

for every m. But a'k~' =a~~>, a and

0'00

are

k-hn+1

perfect, a*=c/>, a~=av ai=a00 , and cf>*=a. Therefore the given conditions are sufficient and necessary for {A } to be p- cgt in a00 --+ cf> 42 ). (X) When a<-=a 00 , F, or Z, the sequence A(m=1, 2, ... ) is p-cgt in a1 --+ a<- if, and only if, (i) ja~~'l <;M for every m, nand k, (ii) the columns of the A are in a<-, and (iii) {Atml} is c-cgt. Proof. The conditions are sufficient. By (i) and (ii), the A are in a 1 --+ a
43 ).

Let x be a point in a 1 , and let y =A x; then

I jxkj 00

y~m> =

.

converges to K (say).

k-1

00

Therefore jy~m'l<

I

00

ja~~'xkJ
k-1

I

k=1

I

00

k-1

jxkj<:MK for every m and n.

a~'xk

(1)

By (i) and (iii), when n is fixed and m increases, the sequence of points [a~':'i, a~':'~,

... ] is a 00 -cgt 44 ). But

a~=a1 ;

therefore lim

m->-oo

00

I

a~'xk

exists

k~1

for every fixed n, i.e., {y} is c-cgt (2) By (1) and (2), {y} is a<--cgt, and hence {A} is p-cgt in a1 --+ a<-. The conditions are necessary. Since {A} is in a1 --+ a<-, the columns of the A are in a<- and the rows are in ai( = a 00 ) 45). Since {A 1m'} is p-cgt in a1 --+ a<-, and in sequence spaces p-convergence implies p-boundedness, therefore {A be the nth row vector of A . Let x be a point in av and let yx; 00

then y~m'= Ia~'xk, and {y}is a<--bd. But a<-<;a00 and a<-*=a~; therefore {ytml} is

k-1

0'00 -

bd. Thus

00

I I a~~'xki < M'

for every m and n. But the rows

k=1

of the A are in 0'00 and a~= a1 ; therefore for every m and n, the sequence of points [a~':'{, a~':'J, ... ]is a00 -bd. Thus ja~~'J, ... , e, . . . are in al" CooKE, (2), p. 326, CooKE, (1), p. 298, of (III) of this paper. 44 ) CooKE; (1), p. 310, 46) CooKE, (2), p. 299, 48 ) COOKE, (1), p. 298, 42 )

' 3)

(6.6, I). (10.4, III); p. 310, (10.6, IV); and following the lines (10.6, IV). (6.4, I). (10.4, III).

BIBLIOTHEEK

MATHEMATISCH AMSnRDAM

CENTRUM

502 Let z=Ae1k>; then z~m>=a~> and {z} is t¥-cgt. Therefore {z} is c-cgt; i.e., lim a~> exists for every n and k; i.e., {A} is c-cgt. m-+oo (XI) The sequence A(m= 1, 2, ... ) is p-cgt in a-+ a, (r> 1) if, and only if, (i) a~>= 0 for every m and n and for k > p, (ii) {A } is c- cgt,

and (iii)

00

~ Ja~>J• < Mk

for every m.

n=l

Proof. The conditions are sufficient. By (iii), the columns of the A are in a,. . . . . . . . . . . . . . . . . . (1) By (1), (i), and (I), the Aim> are in a-+ a,. Let x be a point in a, and let

Y(m)

=

A x, i • e · ' yfl -

Then

Thus

p

a
~ X ,k nk k• k=l

{ J1 IY~m>j• ~ (J 1xkJ")''"..! J1 !a~>j• 1

(2)

p

k=l.

== (a,.k) ;· then

Let c -lim A = A

fl

! Mk k=l

for every m.

lim

=

~ ( ~ !xki'Y1"

m-+00

· { y } Is · c-cgt . I.e.,

y~m>

p

! a.m. xk;

k=l

(3)

By (2) and (3), {y} is a,-cgt 47); and hence {A} is p-cgt in a-+ a,. The conditions are necessary. The fundamental unit vectors are e<1l, e<2 >, ••. , e, ... are in a. Let z=Ae; then z~m>=a~> and {z} is a,-cgt. Therefore lim

a~>

00

!

Ja~>J•.;;;Mk

for every m; also {z} is c-cgt, i.e.,

n=l

exists for every n and k, i.e., {A} is c-cgt. Since {A} is

tn-+00

p- cgt in a -+ a,, and in sequence spaces p-convergence implies pboundedness it follows that {A} is p-bd in a-+ a,. Thus a number p exists such that a~>= 0 for every m and n, and for k > p, by (IV).

Cor. The sequence A(m= 1, 2, ... ) is p-cgt in a,-+ rfo (s> 1) if, and only if, (i) a~>=O for every m and k and for n>p, (ii) {A} is c-cgt,

and (iii)

~ Ja;:z>J• .;;;M,.,

k=l

where

.!:+.!:= 1, r

a

for every m.

Following the lines of (VIII) Cor., the result follows by (XI) and CooKE, (2), p. 326, (6.6, I). (XII)

in

a,-+~¥

47)

When 1¥=G00 , For Z, the sequence A(m=1, 2, ••• ) is p-cgt (r> 1) if, and only if, (i) the columns of the A are in ~¥, (ii)

CooKE, (1), p. 310, (10.6, III).

503

{Alm>} is c-cgt, and (iii)·!

k=l

ja!::1I'
s

Proof. The conditions are sufficient. By (i) and (iii), the A are in a~--+ .x 48). Let x be a point in ar, and let

y

A x; i.e.,

=

00

y~m> =

~a!::> k=l

converges to K (say). Therefore (I)

(

00

00

xk, and

00

~ k=l

jxkir

00

jy~"' 1 1 ~ k~lla!::> xki ~ (k~1 1Xki') 11r (k~1 la!::1 1') 11 ' ~

Kll• M 11• for every m and n.

By (ii) and (iii), when n is fixed and m increases, the sequence of points

[a~"'l, a~':'~• ... ] is a,- cgt 49).

But a:= a.; therefore lim

00

~

m-+oo k=l

a!::>xk exists for every fixed n; i.e.,

{y} is c-cgt. . . . . . . . . . . . . . . . . . (2) By (I) and (2), {y} is a 00 -cgt 50 ). But {y} is in .x and .x*=a!,=a1 ; therefore {y<"'>} is .x-cgt, and hence {A} is p-cgt in a~--+ .x. The conditions are necessary. Since {A} is p-cgt in ar--+ .x, and in sequence spaces p-convergence implies p-boundedness, it follows that {A<"'1} is p-bd in ar--+ .x. Therefore conditions (i) and (iii) are necessary, by (V). Following the lines of (VIII), we can see that condition (ii) is necessary.

Cor.

The sequence A(m=I, 2, ... )is p-cgt in a1 --+a, (s>I) if,

and only if, (i) {A} is c-cgt, and (ii)

00

~ ja~>i•
•-1

for every m and k.

Following the lines of (VIII) Cor., the result follows by (XII) and CooKE, (2). p. 326, (6.6, I). (XIII) When .x = 0'00 , F, or Z, the sequence A (m = I, 2, ... ) is r · p- cgt in a1 --+ .x if, and only if, (i) the columns of the A are in .x, (ii) ja!::11 0 corresponds a number Nk(E) such that ia~-a~jNk. Proof. The conditions are sufficient. By (i) and (ii), and following the lines of theorem (III), we see that the A are in a1 --+ Z. Also by (i) and (ii), the A are in a1 --+ 0'00 61 ), and in a 1 --+ F62). Let x be a point in av and let y =A x; i.e.,

y!a"'1 = 48 ) CooKE, (1), p. (III) of this paper. 49 ) CooKE, (1), p. 50) COOKE, (1), p. n) CooKE, (1), p. 51) COOKE, (1), P·

00

00

~ a~r;:> xk, and ~ ixki is convergent.

k=l

k=l

310. (10.6;) III) ; ·and p. 299, (10.4, V); .see also theorem 310, 310, 298, 310,

(10.6, (10.6, (10.4, (10.6,

III). IV). III). IV).

504

Therefore to

E>

0 corresponds a number t such that 00

I

jxkl ~ E/(4M).

k~t+l

By (ii), Ja;t'l-a~l~la!:'~l+la~~~2M·for everyp,q,n and k. By (iii), when xk#O, to an arbitrary E>O corresponds a number Nk(E) such that ja$]-a~;jNk. Let N'>Nk for k.;;;;t; then I

Jy~v>-y~>i ~) lxda;w-a~~i)l k~ t

·oo

I

+

Jxda;w-a~11)1

k~t+l'

.

00

I

~I lxki·E/(2t·lxki)+2M k~I

E

k~t+I

[xkl ~2

+

2Me 4M

= E for every n, and for p, q > N'. Thus {y} is a 00 -cgt53). But {y} is in IX, and iX*=a~=a1 ; therefore {y} is cx-cgt, and hence {A} is r·p-cgt in a 1 --+ iX. The conditions are necessary. Since the A are in a1 --+ IX, their columns are in iX 54). The fundamental unit vectors e(l), e< 2>, ••• , e=Ae=a~'l:> and {z} is cx-cgt. But iX<;a00 and rX*=a~; therefore {z} is a 00 -cgt. Thus condition (iii) is necessary 63 ). Let a be the nth row vector of A. Let x be a point in a 1 , and let

y =A x; then y<;:'> =

I

00

k=l aDO-

cgt, and hence

a 00 -

and {y} is ex- cgt. Therefore {y} is

a~'l:>xk,

00

II

bd. Thus

But the rows of the A are in a00

m and n, the sequence of points

a~'l:>xkl

< M'

k=l 54 ) and a~

[a~':'l, a~':'i,

for every m, n and k 55).

for every m and n.

= a1 ; therefore for every ja~'l:>i .;;;;M

... ] is a00 -bd. Thus

(XIV) The sequence A(m= l, 2, ... ) is r·p-cgt in 0--+ ar (r> 1) if, and only if, (i) the rows of the A are in a 1, (ii) to an arbitrary E > 0

corresponds a number tk(E) such that arbitrary

E> 0

00

I

n=tk+l

ja~~>ir < E

for every m, (iii) to an

corresponds a number fv( E) such that DO

oo

oo

I I I

n=t,+l

k~p+l

a~'~:>

I< E

for every m, (iv) {A } is c- cgt, and (v) lim I a~~> exists for every n and p. m-oo k=v-1 Proof. The conditions are sufficient. Let x be a point in 0, xk=a for k>p, and let y=Ax; i.e., y~jl

53 ) ' 14 )

55 )

=

l>

I

k=I

a~'l:l

CooKE, (1), p. 314, (10.7, II). CooKE, (2), p. 299, (6.4, I). CooKE, (1), p. 298, (10.4, III).

xk+a

00

I

k~v+l

a~'~:>.

505 Then

1?1.."'11<

p

00

! !a~> x,.! + lal·lk-p+l ! a~'!; 1 ! k-1

p

p

00

< (! jx,.['+ [aj•)ll• (! ja~~>!r +- !

a.~'Z,I')l/r

1

k-1

k=l

=

p

K11r

(! ja~>jr + 1 !

00

k-1

1.·-p+l

a):Zljr)11r,

k-p+1

where

CIO

By (ii), to E > 0 corresponds tk( E) such that ! ja~>j• < E/(2Kp) for n-t'j,+1 every m. By (iii), to E>O corresponds fp(E) such that 00

00

! I ! a~1 1' < E/(2K) fl=fp+l k-p+l for every m. Let N >max (fP, tk) for k.<, p; then 00

!

n-N+l

[y~llr < K (

p

00

!

00

n-N+l k-1

n=N+l k-P+l

< K(2;;K + 2 ~) = E for Let c -lim A = A

lim y~m> = m-+00

p

!

k=l

00

!la~>ir + ! ! !

== (aflk), and let lim m-+00

aJ::ll')

every m. .

00

!

a~>

k=p+l

(1)

= M; then

•

aflk x,.+ aM; i.e., {y1"'1} is c- cgt.

(2)

By (1) and (2), {y} is a,-cgt 56); and hence {A} is r·p-cgt in

C-+ a,. The conditions are necessary.

Since the A are in C-+ a,, their rows are in C*( =a1 ) 57 ). The fundamental unit vectors e(l), e<21 , ••• , e<~<>, ... are in C. Let z=Ae<~<>; then z~m>=a~> and {z} is a;-cgt. Therefore to e > 0 corresponds t,.( E) such that

00

!

n=tk+l

Ia~> I ..;;; E for every m. Also {zlml}

is c-cgt; hence lim a!:f1 exists for every nand k; i.e., {A} is c-cgt. m-+00

The sequence e, in which Let y =A e; then

y~m>

=

e,. = 0 for k..;;; p and e,. = 00

!

a!:f> and {y} is a,- cgt. Therefore to

k=p+l

00

E>O corresponds /P(E) such that Also {y} is c- cgt; hence lim

00

!

57)

CooKE, (1), p. 312, (10.7, I). CooKE, (2), p. 299, (6.4, I).

00

! I !

a~>I'
n=fp+l k=P+l ···

m-+-oo k-p+l

••)

1 for k > p, is in C;

a!:f> exists for every n and p.

506

(XV)

If £X and fJ are perfect sequence spaces, and if the sequence of matrices A} is in fJ* ~£X* 58). Also u'(A_A' 0 corresponds a number N (E, u, X) such that fu' (A
~ E

for p, q > N, and for every x in X.

Therefore, by (1), jx'(A'N, for every x in X, and for a fixed u in (J*. But £X is perfect, hence £X= £X**. Therefore X is p-bd in £X**. Thus {A'
(XVI) If £X and fJ are perfect sequence spaces, and if the sequence of matrices A-£X*. Proof. Since £X and fJ are perfect and {A -AO corresponds a number N(e, X, U) such that ju'(AN. Therefore, by (1), jx'(A'

-A'N. Since £X is perfect, £X=£X**, and therefore X is p-bd in £X**. Thus {A', [. p ·, r · p ·, p · , and p-convergence coincide. Proof. Each of l-p-convergence and r·p-convergence implies pconvergence. Also p-convergence implies [. p ·, and r · p-convergence. Therefore it is sufficient to prove that in a ~ cf>, p-convergence implies p-convergence. 62) If the sequence A, then (i) a~l=O for every m, for k>t, and for n>l, and (ii) {A*(=a); then 58 ) 59 )

60) 61 ) 62 ) 13 )

CooKE, (2), p. 300, (6.4, II). CooKE, (2), p. 284, (6.2, II). COOKE, (2), P· 300, (6.4, II). COOKE, (2), p. 284, (6.2, II). By theorem (VI) of this paper, since {A. Following the lines of theorem (VIII) of this paper.

507

Ixk/ < rk

for every x in X, and Iu,./ < s,. for every u in U; therefore t

!u' (A<:v>_A) xl =I L u,. L n=l I

L

~

xkl

(a~~-a~')J)

k=l

t

for every x in X and u in U.

Ls,.·la~~-a~'iJ/·rk

n=lk=l

Since {A } is c- cgt, to an arbitrary E > 0 corresponds a number such that la$1-a~'iJI ~ Ef(s,. rk tl) for p, q > Nn.k·

N~~>k

Let N'>Nn.k for n
/u' (A<:v> -A)

I

xi~

L

t

L ES,. rkf(s.. rk tl) = E for p, q > N'

.. =tk=l

for every x in X, and every u in U. Therefore {A} is p-cgt in a--+ rf>. (XVIII) In a --+ av l· p., r. p., p., and p-convergence coincide. Proof. As in theorem (XVII) it is sufficient to prove that in a--+ a1 , p-convergence implies p-convergence. If the sequence A ( m = 1, 2, ... ) is p- cgt in a --+ av then (i) a;:;:>= 0 for every m and nand for k>l, (ii) {A<"'>} is c-cgt, and (iii) to an arbitrary

L 00

E>OcorrespondsanumberME)suchthat

n=tk+l

ja~o;;>j

< Eforeverym, by(IX).

Let X be a p-bd set in a, and let U be a p-bd set in ai( =a00 }; then !xk/ < rk for every x in X 64 ), and /uk/ < M for every k and every u in U 65 ). By (iii), to an arbitrary E > 0 corresponds a number ME) such that 00

L

=tk+l

/ai:/:>1 < E/(4Mlrk) for every m. Let A
By (ii), to an arbitrary E>O corresponds a number la~J-a;sJI ~

N~~>k

such that

E/(2flMrk) for p,q > Nn.k•

Let N'>Nn.k for n
I

n=l

k=l

lu' (A<:v>_A) xi= I L u,. L f

~ L

oo

I

~ L

L lun (a$1-a~~) xkl

n=/+1 k=l

l

L /u.. l·rk·E/(2/Mlrk)+

n=l k=l

~

xk/

l

L /un (a$1-a~')J) xkl + L

n=l k=l f

(a~%>-a~'}J)

I

; + L 2MrkE/(4Mlrk) = k=l

z

00

2 2Mrd/a$11+/a~'}JI)

k=l

n=t+l

E for p, q > N',

every x in X and every u in U. Thus {A<"'>} is p-cgt in a--+ a 1 • 64 ) 65 )

(1), p. 300, (10.4, VI). CooKE, (1), p. 298, (10.4, III).

COOKE;

508

Cor. In Goo~~. z,p·., r·p·, p·, ~rul p-convergence coincide. Proof. As in theorem (XVII) it is sufficient to prove that in a00 ~ ~. p-convergence implies p-convergence. Now a00 is perfect, a~= a 1 , and ~*=a; hence if t:he sequence A (m = l, 2, ... ) is p-cgt in a00 4- ~' {A'} is p-cgt in a~ a166). Thus {A'} is p- cgt in a ~ a1 , by (XVIII). But .a and a 1 are perfect; also a*=~ and ai=a00 ; therefore {A} is p- cgt iri G00 ~ ~' by (XVI). (XIX) In ~ ~ a 1 , l· p · , r · p · , p · , and p-convergence coincide. Proof. As in theorem (XVI) it is sufficient to prove that in ~ ~ 1 , p-convergence implies p-convergence. Let the sequence A(m= l, 2, ... ) be p- cgt in ~ ~ ap The fundamental unit vectors eU>, e<2>, ... , e, .. . are in~· Let z=Ae; then z~m>=a~~> and {z} is a1 -cgt. Therefore to an arbitrary E>O corresponds a number ik{E) such that

a

00

.2

ja!:;:>j < E for every m

(l)

n-tk+l

Also lim a!:;:> exists for every nand k; i.e., {A} is c-cgt.

(2)

m-+oo

Let X be a p- bd set in ~. and let U be a p- bd set in ai( = a 00 ) ; then X is of bounded length l, and jxkj < rk for every x in X 67); also jUj,j < M

for every k and every u in U 68). By ( l ), to an arbitrary E > 0 corresponds a number fk( E) such that 00

.2

ja!:;:>j < ef(4Mlrk) for every m. Let /> fk for k.;;;l.

n=fk+l

By (2), to an arbitrary E>O corresponds a number N,.,k(E) such that

Ia$]- a~ I~ ef(2flMrk) for p, q > X,.,k. Let N'>Nn,k for n
iu' (A< 71>-A) xi=

n=l

f

l

00

1.2 u,. .2 (a$1-a~) xki m-1

l

! .

00

~

.2 _2iu,. (a$1-a~) xki + n=t+l .2 k=l .2iu,. (a$]-a~~) xki n=l k=l

~

.2 .21u,.!· rk· E/(2flMrk) + k=ln=f+l .2 .2 Mrdla$Jj + Ja:~ I) n=l k=l

~

I

; +

:

l

l

.22M rk· ef(4Mlrk)

k=l

= E

oo

for p,q > N',

for every x in X and every u in U. Thus {A} is p-cgt in cb 68 )

87) 68 )

CooKE, (2), p. 326, (6.6, I). CooKE, (1), p. 297, (10.4, II). CooKE, (1), p. 298, (10.4, III).

~

a1 •

509

(XX) In a00 "'-+a, l·p·, r·p·, p·, and p-convergence coincide. Proof. As in theorem (XVII). it is sufficient to prove that in a 00 "'-+a, p-convergence implies p-convergence. Let the sequence A(m=l, 2, .. ;) be p- cgt in a 00 "'-+ a. Sinoe a00 is perfect, a~= av and a* = 4>, it follows that {A'} is p-cgt in 4> ~ a1 69 ). Thus {A'} is p-cgt in 4> "'-+ av by (XIX). But 4> and a1 are perfect, 4>*~a, and ai=a00 • Therefore {A} is p-cgt in a00 "'-+ a, by (XVI). (XXI). In a "'-+ a. (s > l ), r · p ·, and p-convergence coincide. Proof. p-convergence implies r · p-convergence. Let the sequence A (m=l, 2, ... ) be r·p-cgt in a"'-+ a,. Since; in sequence spaces, p-convergence implies p-boundedness, it follows that {A } is p- bd in a"'-+ a•. Therefore a~-z> = 0 for every m and n, and for k>l, by (IV). . . . . . . . . . . . . . . (1) The fundamental unit vectors e(l), e<2 >, ... , e, • • • are in a. Let z<•> =A e =a:> and {z} is a8 - cgt. Therefore to an arbitrary E>O corresponds a number fk such that

z Ia:>!•..;;: E for every m. 00

n-tk+l

Also lim a:> exists for every nand k; i.e., {A} is c-cgt .

•

m-+oo

Let X be a p-bd set in a, and let U be a p-bd set in I

I

r

8

-+-= 1; then !xkl ..;;;rk for every x in

X

and

70 ),

a~( =ar),

(2) (3)

where

z lunl' ..;;;M• for every 00

n-l

u in u By (2), to an arbitrary E>O corresponds a number 71 ).

z 00

•-t,.'+l

Ia:>!•..;;: [E/(4lMrk)]• for every m.

f > f~

for k < l; then by (3), a number Nn,k· Let N'>Nn,k ·for n..;;;f and k..;;;l; then Let

such that

/~(E)

exists such that

N~~k

la~-a~l

z ""

•-f+l

ju,. a::J:> xkl ~ rk (

z 00

n=f+l

lunl') 11' (

,! 00

n=t+l

ja~'J:,I") 11" ~ MErkf(4lM rk)

=

E/(4l)

for every x in X, every u in U, and for every m; and thus :u' (ACP>-A
~

n=f+l

~I

z zJu,. t

I

n=l k=l (q)

k-7;:1 u,. ank xk

,__..

(a~-a~)

~ ~ lu..I•E•1"Jo

'''\7::1k7::1

~ ~ £..

£..

2flMrk

for every x in X and every u in U. Hence {A
11 )

z zJu,. oo

CooKE, (2), p. 326, (6.6, I). CooKE, (1}, p. 300, (10.4, VI). CooKE, (1}, p. 299, (10.4, V).

I

n=t+l k=l

~

~

a$J xkl

I

(!>)

+ k-7;:1 n-1+1 u,. ank

(q) u,. ank xk I ~ 2E

k=l n=/+1

18 )

xkl+

El + El4i + 4i =

E

+

xk

I

+

£or p, q > N' ,

510

(XXII) In a,-+ cp (r> 1), l·p·, and p-convergence coincide. Proof. p-convergence implies l·p-convergence. Let the sequence A(m>(m= 1, 2, ... ) be l·p-cgt in a,-+ cp. Since a, and cp are perfect, a:=a,, where~+~= 1, and cfo* =a it follows r

s

that {A'(m>} is r·p-cgt in a-+ a,, by (XV). Thus {A '(m>} is p- cgt in a -+ a8 , by (XXI). But a and a, are perfect, a* =cp, and a:=a,; therefore {A(m>} is p-cgt in a, -+ cp, by (XVI).

REFERENCES l. CooKE, R. G., Infinite matrices and sequence spaces (MacMillan, 1950).

2.

, Linear operators (MacMillan, 1953).