Internat. J. Math. & Math. Sci.
Vol. i0 No. 2 (1987) 227-232
227
.-SIMILAR BASES
MANJUL GUPTA and P.K. KAMTHAN
Department of Mathematics
Indian Institute of Technology
Kanpur, 208016
INDIA
(Received May I, 1986)
ABSTRACT.
(X,T)
locally convex space (l.c.s.)
another l.c.s.
for all
,
Corresponding to an arbitrary sequence space
P
DT
(Y,S)
<--->
is said to be %-similar
{a
if for an arbitrary sequence
{an q(Yn )}
tively the family of all
T
g
%’ for
and S
all q
n
D S, where
{x
a sequence
to a sequence
of scalars, {a
D
and
T
D
which ultimately help to characterize %
and the spaces
s
n
{yn
P(Xn)}
in
E
%
are respec-
T
and
S.
(X,T)
and
(Y,S)
continuous seminorms generating
In.this note we investigate conditions on
in a
n
similarity between two Schauder bases.
We
also determine relationship of this concept with %-bases.
KEY WORDS AND PHRASES. (K)-property, normal topology, semi--base, -base, fully
-base, similar sequences, -similar sequences, normal and symmetric sequence spaces.
gBO AMS SUBJECT CLASSIFICATION CODES. 46A35, 46A45, 46A05.
i.
MOTIVATION.
Similar bases in the literature have been studied with a view to find the compar-
ison between an arbitrary base and a known base of apparently simple looking elements.
However, for certain neutral and obvious reasons, the similarity of bases is defined
in terms of the convergence of the underlying infinite series.
Since there are dif-
ferent modes of convergence, one may talk of similarity of bases depending upon the
particular kind of convergence or a particular type of a Schauder base, e.g. similar
bases, absolutely similar bases etc. (see [I]).
In the literature of Schauder bases,
we have recently introduced different types of
%-bases" which generalize several con-
cepts of absolute bases in a locally convex space (cf. [2]) and hence it becomes imperative to exploit this notion and introduce a kind of similarity, that is, %-similarity
which should have in a natural way some relationship with %-bases and at the same time
should extend the results on absolute similarity.
This is what we have done in this
note and would discuss the same in subsequent pages.
2.
TERMINOLOGY.
Unless specified otherwise, we write throughout
trary Hausdorff locally convex spaces (l.c. TVS)
(X,T)
and
(Y,S)
for two arbi-
containing Schauder bases (S.b.)
228
M. GUPTA and P.K. KAMTHAN
{Xn
E
{Xn ;fn
{yn}
and
{yn;gn}
E
respectively (cf. [3] for details and unexplained
D
We write
terms on basis theory hereafter).
T
continuous seminorms generating the topology
D S.
ing for
isomorphism
(X,T)
The symbol
R
X
from
6
where
p*
E
a Kthe set
{q(yn)}
P, we write
A(P)
and attach a similar mean-
Let
};
x e X
and
T-
family of all
is used to mean the existence of a topological
{{gn(y)}: y e Y}
Pne DT -Wp Ay
x
(X,T)
of
and conversely.
{{fn(X)}:
x
q*
{P(Xn)};
Y
onto
6y
A
(Y,S)
te
to mean
T
A
and
P
DS q,
For
is an arbitrary sequence space (s.s)
For discussion and unexplained
for its KSthe space.
[4]; cf. also [5] whereas for the theory of l.c. TVS our
terms on s.s., we refer to
references are [6] and [7].
An s.s.
A
is said to satisfy the
(K)-propert
if there exists
8
Ax
in
such
that
K
K
E
18nl
inf
8
(2.1)
> 0
A
Without further notice, whenever we consider an s.s.
assumed that
A
as an l.c. TVS, it would be
(A,Ax)
is equipped with its normal topology
Finally, following [8], we recall
DEFINITION 2.2:
if
6X AX,
for each
p
{x ;f
n n
An s.b.
6
(ii) a A-base if
in
D
T’
for an l.c. TVS (X,T) is called (i) a semi-A-base
the map
A
X
X’
and (iii)
(A n(A
(X,T)
p
a fully A-base if
Ix)),
p
X AX
{fn(X)p(Xn)}
(x)
and
is con-
tlnuous.
NOTE:
If
seminorm
If
l
{Xn;fn
Qp,y on
Qp,y(x)
is a semi-l-base, then for each
X
nl Ifn(X) llTnlP(Xn)’
{Qp,y:
D T, there exists
p e
r
in
p(x) _<
3.
in
D
and
T
in
I
x,
the
is well defined, where
{x "f
satisfies (2.1) and
the family
p
n’
DT,Y
D
T
n
e I x}
for all
is a fully l-base, then
of seminorms on
so that for all
nl_ Ifn(X) IP(Xn) -<
x
K8p,8(x)
in
(2.2)
x e X.
T
is also generated by
x; in particular, for each
p
in
X,
_< r(x).
(2.3)
I-SIMILAR BASES.
Let us recall from [I] the following:
DEFINITION 3.1.
{Xn
iliXi
written as
A sequence
{Yn }’
{Xn
if for
converges in
cX
e
X>
is said to be similar to a sequence
{yn
in
i
eiy i converges
in
Y.
The concept of absolute similarity introduced in [I], is abstracted to A-
similarity as follows:
cY,
(3.2)
A-SIMILAR BASES
{yn }c Y,
to a sequence
{enP(Xn)}
X
n
is said to be l-slmilar
{Xn {Yn }’ provided for
{enq(yn )} e l, for all q
to be written as
e l, for all p e D
{x
l, a sequence
For an arbitrary s.s.
DEFINITION 3.3.
229
e D
<--->
T
(3.4)
S.
The following are straightforward.
PROPOSITION 3.6.
{Xn
{Xn
PROPOSITION 3.7.
If
PROPOSITION 3.5.
{p(enXn )}
e
for
{Yn
{Yn
AX
<=>
Ay.
(3.4) is equivalent to
is normal, then
l
e l, for all p e D
<=>
T
{q(nYn)}
e l, for all q e D
(3.8)
s.
in
{x
n
Z1
I
In particular, if
NOTE.
<----->
}l-{yn}
we have
is absolutely similar to
<--->
x
<=>
A(P)
n
{yn
as discussed in
A(Q)
are Kthe
[].
A(P)
A(Q), where
and
spaces corresponding to the KSthe sets
{x
Let
PROPOSITION 3 8
{{P(Xn)}:
p
e D
and
Q
{{q(yn)}:
q
e
respectively.
{yn
and
n
Then
be l-bases.
6
It suffices to observe that
PROOF.
T)
D S}
P
x
A x,
6y
ay
{x
{yn
n
<---> {x
n
{yn }.
and now apply Propositions 3.5
and 3.6.
4. FURTHER CONDITIONS.
In order to obtain more characterizations of l-slmilarlty of bases, let us bringY{x
X and
forth the following conditions for two arbitrary sequences
for all p e D T,
q e D
for all q e D S,
p e D
PROPOSITION 4.3.
l
If
such that
p(xn)
q(yn ),
for all n
Z I.
(4.1)
such that
q(yn)
P(Xn),
for all n
Z I.
(4.2)
S
T
{Yn
n
is normal and conditions
(4.1) and (4.2) are satisfied, then
{Xn ! {yn}
For obtaining the converse of the preceding result, we restrict ourselves to metrlzable l.c. TVS.
DX {Pi:
i
generated by
If
is a metrlzable l.c. TVS, then T
(X,T)
I},
where
Dy
{ql
Similarly, when
P2
Pl
q2 ! "’’}"
is also generated by
(Y,S)
is metrlzable,
S
is
With this background, (4.1) and (4.2) take the fol-
lowing forms:
for all i
i,
for all j > I,
j
and M
i >
and K
i
> 0 such that
pi(Xn)
> 0 such that
qj(Yn
Hereafter, in all subsequent pages, we will let
(2.1).
LEMM 4.6.
let
l
Let
I
I
with
1.(4.4)
for all n > 1.(4.5)
k be a normal s.s., satisfying
x.
(X,T) and (Y,S) be metrlzable.
contain an element
for all n
which will throughout be assumed to be nor-
Also, we consider another s.s.
mal and symmetric along with
<
Miqj(Yn),
Kjpi(xn)
Suppose that
>> 0, that is, a
n
l
is also symmetric and
> 0, for each n
I. Assume
230
M. GUPTA and P.K. KAMTHAN
the truth of the following statement for an
for all i
Then (4.5) holds.
PROOF. Since A x
i,
{pi(nXn)}
A >
E
m:
in
{qj(=nYn)}
,
E
for all
y
is also symmetric, we may find an element
Y >> 0.
Let now (4.5) be not satisfied.
ni(n i
one finds a positive integer
qj(Yni)
>
Then there exists
<
ni+
__i
I)
I, i
in
(4.7)
Jl.
Ax
with
\
so that for each
j
il,
with
pi(Xnl),
for all
yi/qj (Yni)
n
I.
i
Define
d
qj(niYn i)
y
up being
one finds
,
woulb be in
i > m
otherwise
{qj(nYn
and 0 otherwise and so
Yi
,
for otherwise its close
On the other hand, let
a contradiction.
m
I;
then
so that
Pm(niXni ! Pi(niXni
{pm(nXn)}
Hence
i
n
0
Now
n
A
E
for each
I.
m
<
ui’
for all i Z m.
This contradicts (4.7) and the lemma is
proved.
Symmetry considerations lead to
(X,T) and (Y,S) be metrizable, along with A, satisfying the conA
Then {x
{yn} <.> (4.4) and (4.5) hold.
Let
THEOREM 4.8.
ditions laid down in Lemma 4.6.
5.
A-BASES AND
SIMILARITY.
In this section, we investigate conditions when two fully A-bases are similar.
First, we need
LEMMA 5.1.
in
(Y,S).
Let
{x
be a fully A-base for (X,T)
n
{yn
an arbitrary sequence
Then (4.2) is equivalent to the following statement:
for all q
D s,
E
p
{l
for all finite sequences
Let (4.2) be satisfied.
k
q(il iYi
and this proves (5.2).
E
D
such that
T
k
k
q(il iYi
PROOF.
and
!
(5.2)
P(il ixl
k }"
Then lot some r e D
!BB
T
k
il liBilP(Xi)
k
iBB
r(il =ixi
The other part is obvious.
Symmetry considerations in Lemma 5.1 now easily lead to
and {yn
be fully A-bases for m-complete l.c. TVS’s
Let {x
THEOREM 5.3.
and
(Y,S)
n}
respectively.
Then the truth of (4.1) and (4.2) yields that
{x
n
(X,T)
{yn }.
A-SIMILAR BASES
(X,T)
Conversely, if
and
ditions of Lemma 4.6, then
PROOF.
231
(Y,S)
are Frchet spaces with
{Xn
{Yn
l,
satisfying the con-
(4.4) and (4.5) hold.
-->
For the converse,
First part follows from Lemma 5.1 and its symmetrizatlon.
{Xn
observe that
{yn
and
[8].
are h-bases by Propositions 3.5 and 3.6 of
Now
make use of Proposition 3.8 and Theorem 4.8.
Observe that Lemma 5.1 and the first part of Theorem 5.3 and valid for an ar-
REMARKS.
satisfying the K-property.
l
bitrary sequence space
Since h-bases and fully h-bases are the same i a
p.82),
Frchet spaces
we can rephrase Theorem 5.3 for
THEOREM 5.4.
(X,T)
Let
be
Frchet
spaces and
satisfying the conditions of Lemma 4.6.
l,
along with
(Y,S)
and
Frchet
space (cf. [8],
as follows:
{Xn }’ {Yn
Then {Xn
be h-bases
(4.4) and (4.5) hold.
NOTE.
The proof of the above theorem is also immediate from Proposition 3.8 and
Theorem 4.8.
6.
THE ISOMORPHISM THEOREM.
In order to obtain the main result of this section, let us introduce the following
two conditions:
{p(enXn )}
{p(=nXn )}
where
,
e
e l, for all p e D
e l, for all p
{Xn }’ {Yn
D
==>
T
==>
T
{q(enYn)}
{q(enYn)}
DS;
(6.1)
for all q e D S,
(6.2)
e l, for all q
,
X, Y respectively and
are arbitrary sequences in
l,
are
the s.s. as specified in Lemma 4.6.
NOTE.
When
(X,T)
Proposition 4.3).
and
T
are metrizable, the conditions
D
being replaced by
S
DX
and
Dy
(6.1) and (6.2) will
(cf. remarks following
Thus we have
If the hypothesis of Lemma 4.6 holds, then (4.5) --> (6.1) --> (6.2)
PROPOSITION 6.3.
--->
(Y,S)
and
D
be considered with
(4.5).
Next, we have
PROPOSITION 6.4.
ous linear map.
Let
(X,T), (Y,S)
be a metrizable l.c. TVS and
{Xn
For a sequence
X, let
Yn
RXn.
R:X
Y
a continu-
Then either of the equiva-
lent conditions (4.5), (6.1) and (6.2) holds.
PROOF.
in
For each
j > i
there exist
and so (4.5) holds.
X
i
such that
qj(Rx)
KjPi(X)
for all
x
The result now follows from the preceding proposition.
Conversely, we have
PROPOSITION 6.5.
{Xn }’
(Y’S)
Let
(X,T)
be a metrlzable l.c. TVS containing a fully
a Frchet space containing a sequence
{yn }.
h-base
If either of the equiva-
lent conditions (4.5), (6.1) or (6.2) holds, then there exists a continuous linear map
R:(X,T)
PROOF.
(Y,S)
with
RXn Yn’
n _> I.
Define
k
R
then by (4.5) and (2.3),
(m emXm
R:sp {xn
Y
k
m
mYm
is continuous and so its unique extension is
M. GUPTA and P.K. KAMTHAN
232
This completes the proof.
the required continuous linear map.
PROPOSITION 6.6.
{x
(Y,S)
for
{yn
PROOF.
{yn};
with
{yn;gn
cf.
gn
gn(y)
f
(Y,S)
and
(X,T)
be m-complete such that
RXn’
Yn
n
I.
Then
{yn
=
(Y,S). Let
is a semi k-base
{yn }.
n
is clearly an
indeed
X
{x
and
and
(X,T)
be a semi k-base for
n
in
(X,T)
Let
n
{gn
S.b. and let
R-I
n
fn(X)
for all n
I.
Then for each y
Z
Proposition 3.5 of [8].
Thus
corresponding to
X AX
and
(4.2) holds and k
y
x
Y, ther exists a unique
in
is normal,
Consequently, both of these bases are
(Y,S).
is a semi k-base for
Since
i.
s.a.c.f,
be the
Ay.
But
X Y
"->
-bases,
AX
Ay
and now apply Proposition 3.6.
Finally, we have
Let
PROPOSITION 6.7.
{x
k-bases
Rx
with
PROOF.
with
n
and
n
Yn’
{yn
Yn’
be
{yn
n > I.
By Proposition 3.4
Rxn
u-complete barrelled spaces having semi
}. Then (X,T)
(Y,S)
respectively such that {x n
(X,T) and (Y,S)
{x
n
{yn
and so by a theorem of [9], (X,T)
(Y,S)
n> i.
REFERENCES
i.
2.
3.
ARSOVE, M.G.
Similar Bases and Isomorphisms in Frchet Spaces, Math. Ann.
4.
KAMTHAN, P.K. and GUPTA, M.
N.Y., USA, 1981.
5.
RUCKLE, W.H.
1981.
6.
HORVATH,
7.
135
(1958), 283-293.
KALTON, N.J. On Absolute Basis, Math. Ann. 200(1973), 209-225.
KAMTHAN, P.K. and GUPTA, M. Theory of Bases and Concs, Advanced Publishing Program, Pitman, London, U.K., 1985.
Sequence Spaces
and Series; Marcel Dekker, New York,
Sequence Spaces, Advanced Publishing Program,
Pitman, London, U.K.,
J. Topological Vector Spaces and Distributions I; Addison-Wesly, Reading,
Mass., USA, 1966.
KOTHE, G. Topological Vector Spaces I, Springer-Verlag, Berlin, W. Germany, 1969.
k-Bases and Their Applications, J. Math.
8.
KAMTHAN, P.K., GUPTA, M. and SOFI, M.A.
Anal. Appl. 88(I) (1982), 76-99.
9.
JONES, O.T. and RETHERFORD, J.R. On Similar Bases in Barrelled Spaces, Proc.
Amer. Math. Soc. 18(4) (1967), 677-680.