Séminaire d’analyse fonctionnelle
École Polytechnique
R. I. OVSEPIAN
A. P EŁCZY ŃSKI
The existence in every separable Banach space of a fundamental
total and bounded biorthogonal sequence and related constructions
of uniformly bounded orthonormal systems in L2
Séminaire d’analyse fonctionnelle (Polytechnique) (1973-1974), exp. no 20, p. 1-15
<http://www.numdam.org/item?id=SAF_1973-1974____A22_0>
© Séminaire analyse fonctionnelle (dit "Maurey-Schwartz")
(École Polytechnique), 1973-1974, tous droits réservés.
L’accès aux archives du séminaire d’analyse fonctionnelle implique l’accord avec les
conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie
ou impression de ce fichier doit contenir la présente mention de copyright.
Article numérisé dans le cadre du programme
Numérisation de documents anciens mathématiques
http://www.numdam.org/
THE EXISTENCE
iN EVERY SEPARABLE BANACH SPACE
OF
A FUNDAMENTAL TOTAL
AND BOUNDED BIORTHOGONAL SEQUENCE AND RELATED CONSTRUCTIONS
BOUNDED ORTHONORMAL SYSTEMS IN
OF UNIFORMLY
L2
by
R.I.
OVSEPIAN
(Erevan)
and
A.
PE0142CZY0144SKI (Warszawa)
*
Institute of Mathematics of the Armenian
Academy
of Sciences.
#0
Institute of Mathematics,
Polish Academy of Sciences.
Abstract.
1)
*
n
n
(x,)
Tn every
separable
Banach space X
biorthogonal sequence
a
is constructed such that linear combinations of the xl
""
dense in X,
for every
x
in X
if
x
(x)
s
are
n
4~
0 for all
n
then x = 0 and
n
CJ ,
n
11
11
2) Linear subspacps of
which admit
basis consisting of uniformly bounded functions
are
an
orthonormal
characteri-zed.
XX.1
The
present paper consists
In the first
and 4)
we
using
one
of three sections.
trick invented
a
Olevskii (~9~
by
Lemmas 3
prove
Theorem 1 :
In every
Banach space X there exists
separable
a
fundamental
*
and total
biorthogonal
(xn ,xn)
n
sequence
such that
*
Recall that
a
sequence
n
n
of
Banach space X and bounded linear functionals
a
of’
X -
the dual of X is said to be
X, i.e. elements
on
biorthogonal
if
x*(x )
n
~
n,m =
1,2,...
A
biorthogonal
combinations of the
v:’s
are
of elements of
pairs consisting
=
m
amn
for
sequence
(xn ,x n)
dense in
X, and is total if the condition
is fundamental
if linear
-.L
-M-
x*n(x) =
0 for
Theorem 1
1,2, ... implies that
n =
answers
question
a
weaker result has been
previously
([1J, p.238).
of Banach
obtained
by
A
slightly
Davis and Johnson
[4].
The main result of the second section is
Theorem 2 :
L2(03BC)
a
where 03BC
Let E be
is
separable
a
probability
e,
set S. Then E admits
an
bounded functions if and
Moreover if E n
orthonormal basis
can
linear
measure
subspace
on
a
orthonormal basis
a
si g me field
of subsets of
of
consisting
separable subspace
be constructed
space which is dense in the
Hilbert space
a
uniformly
if
only
is
of
norm
11 - 11
so
.0
then the
of
that
it spans
in E n
L-(I,) .
a
linear sub-
XX.2
As
corollary
we
obtain that every
subspace
codimension admits
an
orthonormal basis
consisting
a
infinitely
of H.
many times differentiable functions.
In the third section
an
we
isometric
that there exists
j(X)
ded
a
as
Using
a
question
a
answers
consider the class of such Banach spaces X
embedding,
scalar-valued continuous functions
ball of
uniformly bounded
of
This
of finite
[14J~
Shapiro
which admit
L ~ (0~1)
oi
a
is not
on
a
j,
into
a
space
a
L2(~),
space X has the above
(ii)
satisfies the condition
[13]
result of Rosenthal
i.e. j(X)
11
of all
S such
show that
we
regar-
of Theorem 2.
property if and only if it contains
to the space
of all
S such that the unit
on
probability measure V
totally bounded subset of
L2( )
C(S)
compact Hausdorff space
Borel
subspace of
recent profound
subspace isomorphic
say
a
Banach
closed linear
a
absolutely convergent series
of scalars.
1.
Proof of Theorem 1 .
O1 evski i’ s
tion of
Lemma 3 of
[A]
Banach space X, then
rated
by
Lemma 1 :
We
[9].
a
a
lemma which is
If A is
a
non-empty subset of
xo, x1,...,x
2n-l
2n-1
subspace
Banach space and let
of
n
be elements 0 f X and let
ill,
elements of X
exists
and
then
a
such that
unitary
real
a
subspace of X geneX generated by A.
be a
-
Let
a
denotes the closed linear
A and lin A - the linear
Let X be
modifica-
with
begin
X" 0
positive integer.
-lk
X*1
0 x1,
I
I
*
...,
...7
x 2n_1
x2n-1
bebe.
XX.3
Proof :
The conditions
2x2 -
matrix.
The
(3)
and
(4)
are
specific unitary
satisfied for every unitary
matrix for which
(1)
and
(2)
hold is defined to be the matrix which transform the unit vector basis
2 -dimensional
of the
12-n
Hilbert space
onto the Haar basis of this space.
We put
We have
Clearly (5) implies (1)
Proposition
---- -- -
-
..
---
--
-.
1 :
Let
-.
(x aAl
and
(2).
x 1* )
be
a
fundamental and total
n
I
sequence in
a
Banach
space X
such that there exists
-
nite sequences
(nk)
such that
an
biorthogonal
increasing inf
i-
1,
I
Then there exists
XX.4
a
fundamental and total biorthogonal sequence (e ,e )
n
n
in X such that
iix ))
Proof :
one may assume that
Without loss of
n
generality
f or all
n.
sequence
Next
Pick
(m r)
a
permutation p(.)
of the indices
so
of the indices and
that
an
=
1
increasing
if
put
we
m
akr.
k, j
verify
where
are
defined
as
in Lemma 1
that such defined sequence
for n = m r. Using
*
(e n ,e n)
Lemma 1
has the desired
we
easily
properties.
XX.5
We shall
Proof of Theorem 1 :
of X
implies
assume
that there exist sequences
3~
and
that dim X
F1 C F2 C .M,
of
subspaces
of X
E1 c E2
x =
0.
*M*
In view of
11
Proposition
1
it is
(x) =
enough
~.
C ..,
such that dim
is dense in X and if f
then
=
Then the
of
E~ _
separability
subspaces of X
dim
F~ ;
i for
0 for all
to construct
a
biortho-
XX.6
inductive
hypothesis implies that
and
to
we
a
inf er that
S 3. We define
Thus
linear functional on x.
Remark 1 :
Using
in the
C
and since dim ]
H
H ~F.
to be any
n
3° Day’s technique (cf. [3])
case
the Borsuk antipodal mapping theorem
one
can
choose
(both
g
extension of
prezerving
norm
which bases
in the
case
on
of
*
complex scalars)
real and of
for
s
Now the
1,2,...
=
and x3s
x-
3s
inspection
so
of the
that
proof
separable Banach space for every e&#x3E; 0
and bounded biorthogonal sequence (e
such
n
n
in every
en)
for all
ced
by
n.
1+
However,
c.
1 :
Corollary
norm
We do not know whether for every
III ~ 111
In every
was
separable
,e*)
with
n
by
n
and
sup
lle*ll
n
C.
there exists
yields
a
that lie n nil lien
(1+V2)2 +
n
.
0 this bound
Bessaga
we
be repla-
can
have
equivalent
biorthogonal
an
fundamental and total
1.
n
biorthogonal
sequence in X such
thatile n II
co ..
n
Remark 2 :
A similar
argument
Theorem 1 allows to prove the
that
fundamental
a
Banach space X there exists
=
n
is any fundamental and total
for all
observed
such that there exists in X
(e n
sequence
as
c &#x3E;
of Theorem 1
to that which is used in the
following
proof
of
=
1
c
XX.7
let T :
Theorem 11 : Let X and Y be Banach spaces and
one
bounded linear
operator.
compact, then
*
quences (x n’ ,x n) in X
is not
If X is
separable, T(X)
X -- - Ybe one-to-
is dense in Y and T
there exists fundamental and total
biorthogonal
se-
*
and
(y’ ny)
n
in Y such that
and
for all
2.
n.
Constructions of uniformly bounded
orthonormal sequences.
We
(=
a
employ the following notation. If ~
negative normalized measure) on a sigma
non
.
for any
p-absolutely
and
L2(J1)
is
probability
a
field of subsets of
~
_
as
usually
a
/~
square summable scalar valued functions
denote
measure
x
the Banach spaces of those
and y
on
S.
x
respectively.
The
proof
Instead of
([9J,
Lemma
of Theorem 2 is similar to the
Proposition 1,
a
set S.
product
apply
the
following
of Theorem 1.
result due to
Olevskil
4).
Proposition 2 :
of
we
proof
Let
. , .
Let p be aprobability measure on a sigma field of subsets
be an infinite orthonormal (with respect to the inner
(xn)
n
&#x3E;) sequence of functions in
Then there exists
an
orthonormal sequence
Loo(03BC)
(e n)
such that 1 im inf
such that
n
oo.
XX.8
and
Proposition 2 can be obtained by a non essential modification of the proofs of Lemma 1 and Proposition 1. Actually 01evskil stated Proposition 2 f or the Lebesgue measure on [0,I].
The
of
proof
To prove Theorem 2 it is convenient to
Let
Lemma 2 :
gn
L4(p))
converges to
dimensional
no &#x3E; k
(g n)
and
a
subspace
be
a
use
normalized sequence in
the
following simple
L 2(p)
which
fact.
weaklyY (in
..
zero
Then for every finite
and let
of
say
function h in the
F, and for k&#x3E;0
orthogonal complement
there exist
index
an
of F such that
and
Let
Proof :
20132013201320132013
p = dim F.
Let
l
2
p
any orthonormal basis for F.
’
Pick e&#x3E;0so that
Since
(gI
) conver g es
Clearly
h
belongs
weakl y
to 0 in
L 2(g),
there exists
I
to the
orthogonal complement
of
an
index
n°
and
&#x3E; k
XX.9
We have
and
(i)
It follows from
Proof of Theorem 2 :
that there exists in E
creasing sequence of finite dimensional subspaces
00
U
dim F = p and
~
is dense in E.
F
p
00
subset of
Loo(p)
Loo(03BC)
is dense in E n
there exists in E
In view of
can
one
sequence
a
Proposition 2,
in
(h )
mal sequence
(Fp)
choose the sequence
in the
L°(p)
it
L’(p)
(gn)
is
P E
norm.
is
so
that the
p
a
in-
such that
separable
union U
the
to define
that for
s=
(ii) yields that
assumption of Lemma 2.
inductively
an
orthonor-
1,2, ..,
where I
We define
h1
as
F1
with
the functions
some
the conditions
I,et
any element of
us
consider
(7)
and
separately
~
Suppose
that for
have been defined to
(8) and
two
so
that
cases.
F
The condition
satisfying
enough
so
if
Clearly
F112
C:F2 c ..,.
an
satisfy
XX.10
complete the induction and the proof of the sufficiency
conditions (i) and (ii). The necessity is trivial.
This
A similar
Remark 1 :
Banach space X into
a
and T is not
sup
n
jjx j)
and
Remark 2 :
and
E
by
x2m(t)
F.G.
by
=
0
Arutunian
T(x),x n
denotes the inner
&#x3E;H
It is
2m-1
If E
in X such that
(x )
sequence
n
is
a
an
orthonormal
m=2
enough
where
and
nor
E
H, then
(xn)
biortho g onal
decomposition
E1
for x E X and for
H
admit
of
L 2 0, 1]
(x )
x2
can
sequen-
onto
uniformly bounded
and
to define
n
x.
&#x3E;
of
product
fundamental and total
such that neither
such that the functions
and
a
n n *
orthonormal bases.
2
T(X).
operator from
is separable
#
is defined
There exists
E1
bounded linear
orthonormal basis f or E. Moreover if X is
n
.,
E2 =
an
*
E X
n=1,2..., where . , .
be chosen so that (x
subspaces
is
one
Let E =
Hilbert space H.
a
(T(x ) )
*
and
to
one
a
then there exists
compact
oo
separable
argument gives
Let T : X -- -~H be
Theorem 2’ :
of the
is any orthonormal basis for
are
unbounded,
(m= 19 2y
(unpublished) we have
However
x 2m-1 0
as
was
for
0t2
observed earlier
XX.11
Corollary 2 : If E i s a linear subspace of a separable
where p is a non purely atomic probability measure and
complement
of E is finite
orthonormal baxis.
can
a
Moreover if E --,
be chosen from elements of E Q
Proof :
It
is
to show that
enough
space
the
L-(J.1)
orthogonal
uniformly bounded
in E then the basis
is
L~(~.) .
rE]
(ii) of Theorem 2 To check (i) first observe that the density
r
regarded
integer
as
L2(~)
of
subspace
a
p and for every
in
L2( )
implies
linearly independent
"
00
there exist
(i)
satisfies the conditions
of
that for every
f1,f9,...,f
2
1
.1
p+1
and
L Ð (~)
positive
in
such that the matrix
is invertible.
Let
for i =
1, 2, ... , p+1.
thogonal complement
existence of
to any basis of the or-
The above observation
applied
of E and any
element f of
y in
an
be the inverse matrix and let
i,k
L 00 (J.1)
non
such that
zero
y~f&#x3E; ~
1 and
[E] yields
the
~r, g &#x3E; ~ 0 for all g
orthogonal complement of E. The last condition means that y E
Hence there is no f/0 in [E] which is orthogonal to all y E
Lco(g)
is dense in [EJ- Hence (E] satisfies (i).
equivalently
in the
The"moreover" part of the Corollary follows from the observation
that if
[E]
satisfies
(ii)
than E also satisfies
An immediate consequence of
Corollary 3 :
(ii).
Corollary 2 is
Let f be an y unbounded function in
Then the ortho-
uniformly bounded orthonormal basis consisting of trigonometrical polynomials. This basis has no extension to any
uniformly bounded orthonormal basis for
gonal complement
Corollary
of f admits a
3 answers
a
question of Shapiro
[l4]
XX.12
Fat
3.
Definition :
space S.
to
a
A closed linear
a
spaces.
subspace
Loop) - L2(~)
I
fat with
:
or
Z of
regarded
as
measure
C(S)
a
Hausdorff
compact
a
on
is said to be fat with
respect
L2( )
subset of the Hilbert space
denote the natural
respect to p iff
operator
Borel
probability
bounded set.
totally
Let
rem
be
Let 03BC
C(S)
if the unit ball of Z
is not
Z is
of
subspaces
equivalently
injection.
A
It is clear that
the rest.riction of I
if E T I
to Z is not
p
satisfies the condition (ii)
(Z)
compact
a
of Theo-
2n
Our next result characterizes Banach spaces which admit fat isometric
embeddings
in terms of
C(S) spaces. Some of the equivalent conditions are stated
2-absolutely summing operators, i.e. such bounded linear ope-
into
rators which admit
C12’
measure (cf.
factorization
a
and
through
a
equivalent :
(a) there exists
such that
injection
follwing
conditions
uniformly bounded sequence (x n) of elements
subsequence of (x n) is a weak Cauchy sequence,
no
there exists
a
2-absolutely summing operator from
(d)
there exists
a
2-absolutely summing
y
non
X onto
1~
y
compact operator from
X
12,
y
for every for
there exists
is fat with
Proof :
to
of X
11,
(c)
subspace isomorphic
a
some
are
a
X contains
(e)
for
p
e
(b)
into
I
[8]).
For every Banach space X the
Proposition 3 :
natural
some
a
probability
Let T ba
embedding j
of X into
Borel measure 03BC
on
a
C(S)
space
j(X)
S such that
to p.
respect
(a) ~ (b). This is
(b) ~ (c).
isometric
profound
a
a
recent result of Rosenthal
bounded linear
operator from
11
[13].
onto
12
XX.13
(cf. ~2~
for the existence of such
~7] (cf.
Grothendieck
T admits
extension to
an
a
T is
Then
by
a
result of
2-absolutely summing. Hence, by
2-absolutely summing operator from
I)
(c) = (d).
Obvious.
(d) ~ (e).
Let T :
operator
and let S be
[11],
and Pietsch
exists
a
Borel
image
j(X)
(e) =~ (a).
(x n)
A :
L 2(p) __ .12.
of j(X)
is
under
(x n) does
Cauchy
(e)
with
n
uniformly
a
1 for
Cauchy sequences
strong Cauchy sequences.
3
was
subset
bounded
n/m (n,m=1, 2,...)
not contain weak
Proposition
the
respect to .
2
m
some
because
recently independently
by Weis [16J.
discovered
Our last result is related to
result of Sidon
Corollary 4 :
tends
a
that there exists
sequences into
our
is not
Ip
A l 03BC j for
is not compact,
totally bounded
Since T
subspace of C(S)
fat
a
A similar result to
Let
result of Persson
a
S such that T=
on
2
in X such that
takes weak
compact 2-absolutely summing
non
2
It follows from
Thus the sequence
of S.
a
for every isometric embedding j : X -- -~ C( S) there
operator
Thus
sequence
be
compact Housdorff space. By
a
of the unit ball
of
-- ~12
X
probability measure p
bounded linear
-vjea.’,,-.1y
to
infinite
of
a
[15].
I,et p
( gn)
Gaposhkin’s [6] generalization
be
a
zero
be
a
probability
uniformly
in
L2(p)
measure
on
a
sigme
bounded sequence in
Loo(J.1)
and lim sup
0.
field of subsets
such that
subsequence
and c &#x3E; 0 such that
nk
for every finite sequence of scalars
(gn)
Then there exists
n
an
X
12.
onto
I f.1
F8])
also
operator).
an
c ,c ,...,c (p~l,2,...).
XX.14
Without loss of
Proof :
Then
(g)
weakly
does not have
conver g es in
converges to
zero.
generality
Cauchy (in
L2(p)
Thus
g~ ~
regarded
L ( ~) ---L 2(p)
inj ect ion
L 2(p))
but
zero
as
that
a
because
subsequences
subse q uence
no
of
(g )
(g )
strongly
L’(p)
sequence of elements of
sequences because the natural
takes weak
in
stron g Cauchy sequences
assume
may
Lm(p))
does not contain weak
into
to
we
L 2(p).
Cauchy
sequences in
Since su p
complete
n
the 1 -,,,roof
it is
for the real
enough
case
to
and Dor
apply Rosenthal’s
[5]
criterion
(cf. Rosenthal
[13]
complex case).
for the
***
*
REFERENCES
Théorie des opérations
Warszawa 1932.
[1]
S. BANACH :
[2]
S.
[3]
M.M.
[4]
W.J. DAVIS and W.B. JOHNSON :
[5]
L.
[6]
V.F.
[7]
A. GROTHENDIECK :
BANACH und S. MAZUR :
DAY :
DOR :
linéaires, Monografie Mat.,
Zur Theorie der linearen Dimension,
Studia Math. 4 (1933), 100 - 112.
On the basis problem in normed linear spaces,
Amer. Math. Soc. 13 (1962), 655- 658.
On sequences
GAPOSHKIN :
spanning
On the existence of fundamental
and total bounded biorthogonal
systems in Banach spaces, Studia Math.
45 (1973), 173-179.
a
complex
l1
J.
space,
to appear.
Lacunar series and independent functions,
Nauk, 21 (132) (1966), 3 - 82 (Russian).
Résumé de la théorie
LINDENSTRAUSS and A.
PE0142CZY0144sKI :
Usp.
Mat.
des produits tenSoc. Matem. Sao Paulo
métrique
soriels topologiques, Bol.
8 (1956), 1 - 79.
[8]
Proc.
Absolutely summing operators
in Lp spaces and their applications, Studia Math. 29 (1968),
275-326.
[9]
A.
M.
OLEVSKI~ :
Fourier series of continuous functions with respect
to bounded orthonormal systems, Izv. Akad. Nauk SSSR,
Ser. Mat. 30 (1966), 387-432 (Russian).
XX.15
[10]
A.
PE0142cZYNSKI :
[11]
A.
PERSSON und A.
A note on the paper of I.
and reflexivity of Banach
21 (1962), 371 - 374.
PIETSCH :
Singer "Basic sequences
spaces", Studia Math.
p-nukleare und p-integrale Abbildungen
in Banachräumen, Studia Math. 33 (1969),
19-62.
[12]
A.
PIETSCH :
Absolutely p-summierende Abbildungen
Math, 28 (1967), 333 - 353.
in normirten Räumen,
Studia
[13]
H.P. ROSENTHAL :
[14]
H.S.
SHAPIRO :
A characterization of Banach spaces
Proc. Nat. Acad. Sci. USA to appear.
containing
l1,
Incomplete orthogonal families and related question
orthogonal matrices, Michigan J. Math. 11 (1964),
on
15 - 18.
[15]
S.
SIDON :
Über
orthogonalen Entwicklungen,
(1943),
[16]
L.
WEISS :
On
in
Acta Math.
Szeged
10
206 - 253.
strictly singular
preparation.
and
strictly cosingular operators