I ntrnat. J. Math. & Math. Sci.
Vol. 6 No. 3 (1983) 431-447
431
ON THE LATTICE THEORY OF FUNCTION SEMI-NORMS
I.E. SCHOCHETMAN
and
S.K. TSUI
Department of Mathematical Sciences
Oakland University
Rochester, Michigan
48063
(Received October 11, 1982)
ABSTRACT.
We consider the lattice of function semi-norms over a measure space,
as well as certain distinguished subsets.
We determine which subsets are
sublattices and, in turn, which of these are Dedekind complete.
We also investigate
the extent to which the distributive and DeMorgan Laws are valid in this setting.
KEY WORDS AND PHRASES. Function semi-norm, associate semi-norm, infimum,
spremum, Riesz-Fisher propey, strong and weak Fatou properties, infinite
tiangle inequality, absolutely continuous norm, (sub-) lattice, Dedekind
completeness, distributive la DeMorgan s Laws.
1980 AMS MATHEMATICS SUBJECT CLASSIFICATION CODES.
Secondary: 06A40, 46B05
1.
Primary:
06A23, 06A35, 46E30
INTRODUCTION.
In this paper, we investigate the sublattice structure of the lattice of
semi-norms on a fixed measure space.
Section 2 is devoted to establishing the necessary preliminaries.
In
Section 3, we distinguish the semi-norms of interest, namely those having either
the Riesz-Fisher, weak or strong Fatou properties, those satisfying the infinite
triangle inequality and those which are of absolutely continuous norm.
In
determining the sublattice status of each of these collections, we are also
interested in Dedekind completeness for each of these, as well as for the space
of all semi-norms.
In particular, when a subset is closed under arbitrary
suprema or infima, we wish to know the specific description of these extrema.
The answers to these questions in the case of the supremum are essentially known.
I.E. SCHOCHETMAN and S. K. TSUI
432
For the sake of completeness, these are summarized in Section 4.
However, the
answers to these questions in the case of the infimum are hardly known at all.
In
fact, the notion of infimum is not unique; it depends on which sublattice is
being considered.
The main difficulty here involves the infimum of an arbitrary
collection of function semi-norms.
There are three candidates for this notion
Here we also determine which infimum
which we define and study in Section 5.
goes with which sublattice.
In Section 6,
This is the main section of this paper.
we are then able to determine (i) whether a given subset of semi-norms is a
sublattice and (2) if it is complete.
In Section 7, we turn our attention to the DeMorgan Laws.
As we shall see,
the validity of these laws depends on which of the two laws is being considered--as
well as its interpretation.
In concluding, Section 8 observes that the sublattices under consideration
are not distributive in general and also shows that there is a connection between
the lattice theory of function semi-norms and finite sums of such semi-norms.
2.
PRELIMINARIES.
Here we recall the preliminary results we will require throughout the
A general reference for this section is [5].
remainder of this paper.
Let
(X,S,)
As usual, the real numbers and the natural numbers
non-negative real numbers.
will be denoted by
If
C
[0, =] the extended,
be a fixed sigma-finite measure space and
and
is the complex
equipped with their usual measure structures.
numbers, define
M
{f:X
C
f
-measurable}
is
and
+
M
{f:X
[0, ]
f
g-measurable}
is
In general, we will not distinguish between elements of
M
or
+
M
which are
equal almost everywhere.
In will also be convenient to denote
DEFINITION 2.1.
satisfying:
x
A(function) semi-norm on
X
f(x) # 0
+
M
by supp(f)
is a mapping
p
+
M
[0, ]
433
LATTICE THEORY OF FUNCTION SEMI-NORMS
o(o)
(i)
0
o(af)
(ii)
a 0 (f)
0
a
o(f + g)-< p(f) + o(g)
(iii)
o(f) -< o(g)
(iv)
The semi-norm
0
0
II’II P
f _< g
P
Also, for
f
in
0
Clearly,
LEMMA 2.3.
Then
(i)
(ii)
g
For
f
is the
P
Let
0
denote the
Of course,
(f)
+
02(f
f s M
(f)
define
0
P
Then
e(0)
f # 0
0
Then
are the smallest and largest elements of
Let
P
0
and
/
g s M
Og(f)
Define
P
o
P
e
o
P
+
o(fg)
f s M
Of particular interest are the cases where
is the characteristic function
p
_>
Ol(f)
if
M
in
define
is a semi-norm.
g
f
implies
is canonically partially ordered by
+
M
e
and
+
M
_< p _<
Pl >- P2
EXAMPLE 2.2.
0
g
the subset of all norms (never empty).
o
is a norm for each
Observe that
f,g
o(f)
is a norm if
P
+
M
f,g
whenever
set of all semi-norms and
+
M
f
X
E
of a measurable set
E
or
El-norm [[’[[I
Now define
LO
{f s M
NO
{f s M
o([fl)
}
<
and
Then
L0
LP
@(Ifl)
N
is a semi-normed linear space with
[O/N@
a closed subspace.
II’]I
If
is a normed linear space.
0}
L0
then
P
Thus,
is the familiar
LP-space
EXAMPLE 2.4.
so that
LO__c
OE
L
Let
g
X
as in
E
2.3, with
On the other hand, for
in general,
LP=A
{L
DE
E
@
X
P
Then
we have
PE
L0
Og -<
OE
L
O
Hence,
I.E. SCHOCHETMAN and S. K. TSUI
434
REMARK 2.5.
pi,2 belongs
If
P
to
DEFINITION 2.6.
then the pointwise supremum
sup(pl:02)
of
[5,p.442].
This is not the case for the pointwise infimum.
Let
s P
i’ PZ
inf{@l(fl)
inf(ol’@2)(f)
The function
P
02’ 02
inf(Pl,@2)
For
+
f
02(f2)
+
M
in
define
M+’fl + f2
fl’f2
is a semi-norm and is the infimum of
f}
P
in
i, 2
[5,Ex. 71].
In what follows, it will be convenient to write
and
A
Pl
for
2
+
[o,,]
F-- {:M
F
’
define the associate
v
(0) --o)
For such
need not be a semi-norm.)
Sup{/x
+
@(g) _< I}
(fg)d
f s M
Observe that the supremum is taken over a non-empty set.
p’ s P
LEMMA 2.7.
"
then
LEMMA 2.8.
(Pl
A
02
Pl
PROOF.
’0p
The function
(@p)’
and
0
is a
_< p _<
(i)
If
_< p
and
g
01,02
p
(n)
P
p
and
P2
(I
v
i -< 2
and
(n+2)
(Finite DeMorgan Laws)
V
co, co’
0’
In particular,
I/p + I/p’
where
we may
p
by
@’(f)
semi-norm, i e
sup(o1,P2)
for
2
Now let
inf(01,02)
(Note that an element of
@I
2 )’
Pl
The first law is well-own
i’ 2
A
(ii)
2
Pl
If
1,2,3,
n
If
then
s P
then
P2
[5,Ex.71.2].
The second law is much
less well-known; in fact, there appears to be no proof of it in the literature.
The proof we give next is due to Luxemburg [3].
We are grateful to Professor
Luxemburg for permitting its inclusion here.
01,02
g
P
we have
(I’ ^ )
>-
(01
v
For
i.e.,
and suppose
is trivial)
f
I’02
02 )’
Since
f
I v 02
so that
01 02
To prove the other inequality, let
is an element of
L 0’
<
+
M
such that
0’
(f) <
(
v
02
01
(i 14),
v
02
(otherwise, equality
it corresponds to a bounded linear functional
435
LATTICE THEORY OF FUNCTION SEMI-NORMS
L
on
p
given by
g s L
fgd
(g)
X
Also, the Banach dual norm
from
L
Pl
[2, p. 247]
L
P2
p*
of
p’
is equal to
p
L
on
p’
It then follows
that there exist non-negative linear functionals
I+2
such that
p’(f)
L
on
p
L
on
p
on
2
and
+
pl(l
p (@)
p (f)
i
P2(2
it follows from the Radon-Nikodym Theorem that
Since
i, 2
i,@ 2
are integrals, i.e., there exist functions
fl’ f2
0
in
L
Pl
pV
L
2
respectively such that
/
i(g
p,
Thus,
fi
s L
3.
A
Pi’(fi)
Pi*(i
and
fl + f2
f
follows that
P’ (f) >- (Pl
i
P2’)(f)
p’(f)
also,
(Pl
i.e.
i
V
p2 )’
L
g
gfi d
X
1,2
A
I+2
Since
pl(fl)
Pl
p
+
it
Consequently,
P2(f2)
P2
THE SUBSETS.
P
In addition to the subset
P
chain of subsets of
of
F
we will be interested in studying a
having certain desirable properties.
In this section we
recall these properties, show their relationships to each other and consider some
References for this section are
general examples.
[4,5].
With the exception of
the subset satisfying the infinite triangle inequality, these subsets have been
studied before.
DEFINITION 3. i.
(i)
p
P
Then:
has the Riesz-Fisher Property (RFP) if for each sequence
p
such that
%p(f
such that
p(fn-gn)
n
it follows that there exists a sequence
<
0
and
{f
n
{gn }
in
in
+
M
+
M
p(Egn) <
satisfies the Infinite Triangle Inequality (ITI) if for each sequence
(ii)
p
{fn
in
(iii)
Let
0
+
M
it follows that
p(Efn -< Y’P(fn
has the Weak Fatou Property (WFP) if for each sequence
{fn }
in
+
M
I.E. SCHOCHETMAN and S. K. TSUI
436
such that
(iv)
f
such that
p
i f s M
f
n
+
f
+
M
f
in
fl
>_ f
it follows that
<
it follows that
is of Absolutely Continuous Norm
to be the subspace of
f
p(fn
and lim
p(f) <
has the Strong Fatou Property (SFP) if for each sequence
O
(v)
+
n
L
n
f
L
in
p
(fn)
lim
(ACN) if
L
Let
0
the sequence
{(f n )}
R (resp.l,W,S) denote the subset of
(resp. ITI, WFP, SFP) semi-norms.
P
subset of
B
let
R ,I
the subsets
o
o
o
P
A
Also, let
denote the set
,Wo,So,Ao
L
where
P
of
B
P
in
o
Recall that
such that
consisting of the RFP
denote the subset of
B
Finally, if
B
of norms in
o
I
where
o
+
M
is defined
a
0
W) consisting of the ACN semi-norms.
(equivalently
L
a
{f }
n
converges to
M+
in
(f)
p
having absolutely continuous norm.
has this property if for each sequence
+
{f }
n
o
is any
Thus, we obtain
[4, Thm.4.2].
R
S
We then have
the following sequences of proper inclusions:
A c S c W c I c R c P
Moreover,
B
s A
and
o
c B
o
.)
Ao c S o c Wo c R c P
o
o
and
for each choice of
B
e A
(Note that
A,S,W,I,R,P
These inclusions, as well as the counter-examples for properness,
can be found amongst the results and examples of [4.5].
LEMMA 3 2
If
pg
is as in
pg
2.3, then
S (resp.A) if
belongs to
does.
Recall that if
LEMMA 3.3.
PROOF.
S
since
then
s F
then
If
o--II.II
Let
such that
O s F
Then
@(g) _< 1(3.2)
P
and
S
’
’
s P; actually, more is true.
s S
Also, if
o s S
and hence,
Moreover,
’
sup{o
p s S
o
g
g
s S
then
"
for all
o(g) _< I}.
o
g
in
Hence,
are closed under pointwise suprema (left to the reader).
The second part of the lemma is well-known.
QUESTION 3.4.
is it automatically
4.
Recall that
WFP, i.e.,
A
in
{p s W
p is ACN}
P
is
ACN,
W ?
SUPREMA.
We begin our study of the lattice theory of
P
If
P
by asking if the subsets of
introduced in Section 2 are closed under the formation of arbitrary pointwise
LATTICE THEORY OF FUNCTION SEMI-NORMS
We have already seen that
suprema.
P
have this property" (proof of 2.3).
S
and
437
Our objective here is to summarize (for the sake of completeness) the known answers
to this question for the other subsets.
P
4. i.
{pj
fact, if
norm.)
is a
j}
j
Pk
is a
norm, for some
are just semi-norms [4].
@j
0
R
all
I
and
o
J
j
W
j
k
sup(j)
to be a
sup(@j)
All that is required of the
0.
R [4,p.147].
are closed under the formation of finite pointwise suprema.
o
This is not true for infinite collections.
W
sequence in
Although
In fact, there exists [4,p.150] a
W
whose pointwise supremum is not in
o
I
R
o
we have seen that
o
W c I c R
I
Furthermore,
closed under the formation of arbitrary pointwise suprema, while
not.
then
are closed under the formation of arbitrary pointwise suprema.
W
and
f
implies
This is not the case for
4.3
and
for their pointwise supremum to be a norm is that they be total, i.e.,
pj(f)
4.2.
c__ p
The converse is false, i.e., it is possible for
norm while all the
Oj
(In
is closed under the formation of arbitrary pointwise suprema.
o
W
and
is
R
are
This is good reason for distinguishing the ITI property from WFP and RFP in
general.
4.4.
S
(as well as
o
is closed under the formation of arbitrary pointwise
S
suprema.
4.5.
A
and
Ao
are closed under the formation of finite pointwise suprema.
This is not true for infinite collections.
A
o
whose pointwise supremum is not in
QUESTION 4.6.
5.
Is
R
In fact, there exists a sequence in
A
closed under the formation of finite pointwise suprema?
INFIMA.
In Section i, we saw that the infimum in
P
of two semi-norms
is
@i,2
given by
(I
A
O2)(f
inf{Pl(fl)
+
2(f2)
fl,f2
+
M
fl + f2
f}’
+
f e M
There are two cononical generalizations of this formula to the context of an
arbitrary collection
{pj
j
J}
of semi-norms.
In this regard, it will be
I.E. SCHOCHETMAN and S. K. TSUI
438
convenient to refer to a collection
+
M
in
(i)
(of countable type)
0
fj
If
f
as a J-decomposition of
f
j
J
in
in the almost-everywhere sense.
(i’)
has the property
(fj)j
+
M
in
if:
for all but countably many
Z f.
J
j
(ii)
(fj)j
0
f.
then we will say it is of finite type.
k(f)
inf{Z
pj(fj)
J
for all but finitely many
+
Now let
f s M
j
J
in
and define:
is a J-decomposition of
(f)
is a J-decomposition of
f}
(fj)j
and
inf{Z p.(f
y(f)
J
J
3
j
J
f of finite type}
Note that the sums in each case have at most countably many non-zero terms.
As suggested by Luxemburg in [3], there is another less canonical generalization
(Pl ^ 2 )(f)
of
I’ v P2’
(I ^ 2 )’
motivated by
namely,
(sup(pj))’
o
In this section, we shall make use of all three of these generalizations.
It is well-known that
THEOREM 5.1.
PROOF.
o
is a semi-norm; in fact,
Suppose
fl
fk
g
f
+
M
f, g
fl
Define
f
f _< g
with
Then
k
f
and
n
+
+
fn+l min(fngn+l) Then
fn’ i.e.,
fn+l -< f’n f fl
{fk
obtain a sequence
Ef
We have
k
f
in
fn -<
fn+l -<
f
+ such
M
+
that
needing verification is the
fl
f
is similar to that of
y
{gk}
Let
min(f’gl)
Define
X
k
The proof for
Suppose we have obtained
0
I-
gk
next show
is a function semi-norm.
The only part of the theorem for
monotonicity; we will show this.
Zg k
y,k
Each of the mappings
Moreover:
S
o
be a sequence in
-< gl
and
1’’’’’
f
f
n
so that
gn+l
+
fk
fl
-<
f
in
M+
f’n
f
+
M
k
such that
i.e.,
such that
fl
f
n
>_0
and
fn + fn+l
f
and
<-
In this way, we
f
fk
gk
Let
x
k
We
(in the almost-every-where sense).
f(x) _< g(x)
outside the null set.
for
If
x
f(x)
outside a null set.
g(x)
then
fk(x)
gk(x)
be a point in
for all
k
so
439
LATTICE THEORY OF FUNCTION SEMI-NORMS
that
Efk(x
f(x)
If
g(x)
f(x) < g(x)
then
g(x)
%gk(x)
f(x)
gi(x)
fi(x)
and hence,
for some
i
Consequently,
f’
fi(x)
i-I
(x)
f(x)
fl(x)
+
fl(x)
fl(x)
fi-l(X)
i.e.,
f(x)
Zf
which implies
k(x)
f(x)
+
Zf
i.e.,
Therefore, each decomposition of
(fk)
a sequence decomposition
-<
of
+ having
M
in
+ yields
M
in
(gj)j
is a J-decomposition of
_<
k(g)
j)j
f
J-decomposition
f
of
X,T,o
y,k,o
T
k
LEMMA 5.2.
If
p
j e
J}
{@j
+
f s M
Let
PROOF.
<_ g_.
fj
{pj
Necessarily,
J}
j
k _< T
Obviously,
p
I)
J
Moreover, if
and
p
is
is a lower
X.
_<
be a J-decomposition of
(fj) J
and let
j e J
is over a larger index set.
satisfies the ITI (i.e.,
then
Hence,
there exists a corresponding
for which
i(f)
g
0)
g_.
for
is a lower bound for
Next we wish to compare
finite, then
+
M
in
in
since the infimum for
Observe that each
bound for
g
<- gk
+
in M
fk
the properties
0
f.
since such is essentially a sequence (define
k(f)
(gk)
by a sequence
Consequently, the same is true for any J-decomposition of
k
if
fi(x)
f
k
g
f
+
fi_l(X)
f
in
+
M
By our hypothesis, we have
o(f)
p(Z
J
o < k < y
Then
PROOF.
Hence,
o
o"
is an upper bound for
Consequently,
%’"
_<
k"
o"
S c I
and
o _<
%’"
k"
o"
-<
@(fj)
@j(fj)
%
J
be a subset of
T"
o
o
Since
j e J}
{pj
Let
and
%
J
p(f) _< X(f)
from which follows that
THEOREM 5.3.
-<
fj)
{
o
Now
%" _<
j
J}
and
o
pj
implies
Thus,
%’"
J
j
@j
P
with
_<
X _< %"
j
J
ie
since
’
we have
%’’ > Oj
%’’ >- o’
,k,o as above.
(5.2)
%,’
sup(p)
This establishes the equalities.
The
440
I.E. SCHOCHETMAN and S. K. TSUI
following examples establish the inequalities.
EXAMPLE 5.4.
X-- IN
Let
Define
Zf(n)/2
(f)
n
lim sup f(n)
(f)
and
(f) + (f)
pj(f)
{pj}
so that
X
EXAMPLE 5.5.
p"
we have
Let
"
< p--
@"
X, i.e.,
S
such that
O
{Oj}
{O,2p}
For the collection
<
X
o <
For finite collections in
LEMMA 5.6.
P
be a non-trivial element of
p
in which case
#
o
We leave to the reader the task
{@j}
for this choice of
< y
1,2,
j
is a constant sequence of norms.
of verifying that
i.e.,
+
M
f
S
X =y (use 2 .8)
we have
For the remainder of this section, we investigate the status of the infimum
in
y
Once again, for
LEMMA 5.7.
The infimum of the
_
In particular,
infp(@j)
.
{@j
P,R,I,W,S,A.
A
i
j
J}
!
in
P
is
P
let
be as above.
notationally,
T
In this sense,
infp(Pl,02
2
a,X,y
P
is
closed under the formation of arbitrary infima.
LEMMA 5.8.
The infimum of a family in
P
also belongs to
example, consider
THEOREM 5.9.
notationally,
I
sense,
Z(gn) <
This is not true for arbitrary families in
o
{(I/n)}
{j}
If
infl(Pj)
P
for
I
X
J
n
o
P
P
o
(For
o
.)
then the infimum of the
In particular,
Fix
> 0
{gn }
and let
A
Pl
2
be a sequence in
By the definition of
pj
in
I
is
In this
infl(Ol’@2)
of
gn
in
for each
{j
jn
+
M
for which
there exists a
n
jn-decomposition
+ such that
M
k(gn
jnZ oj(gn,j) <
Let
which is bounded below in
is closed under the formation of arbitrary infima.
PROOF.
(gn,j)jn
Po
gn,j
# 0}
oj(gn,j)
Then
Z
J
J
n
is a countable subset of
@j(gnj)
n
+ s/2n
n
1,2
jn
and
441
LATTICE THEORY OF FUNCTION SEMI-NORMS
Let
j
UJ
J
J
n
then
which is also a countable subset of
0
gn,j
Z Z
n J
P
all
J (gn
Z
<
Furthermore, if
Observe that
n
j
UJn
[X(gn)
+ e/2 n
k(gn)
+ e
n
n
_< Z
n
Therefore, rearranging the double series, we obtain
gn,
pj()
Z Z
n J
Z Z
n J
n
% %
J n
Moreover
{gn j
j
Z Z
n J
Thus,
{gn,j
g
J
Z
Pj(gn J)
IN
n
gn ’J
pj(gn j)
+
M
is a countable subset of
satisfying
gn
+
M
is a subset of
IN x J
which is indexed by
has countably
many elements and satisfies
Z Z
Jn
Thus, the family
(Z
n
gn
Z Z
gn,j
nJ
gn,j)j
Z
n
gn
is a J-decomposition of
gn
Z
n
Consequently,
k(Egn)
n
_<
Z
J
@j(Zg
j)
n n’
Z Z
@j(gn j)
J n
-< E Z @j (gn
n J
(@j
s I)
j)
n
< Z
n
k(gn
+
,
for arbitrary
> 0
is
+
M
442
I.E. SCHOCHETMAN and S. K. TSUI
Hence,
n
k
which implies that
p
If
Hence,
k
I
I
is in
{Oj _c Ro
If
COROLLARY 5.11.
is the infimum of
W
is
PROOF.
If
If
W
and
Pl
e W
pl,O2
W
0 <
1,2
Letting
i
1,2
From this it follows that
min(al,a 2)
a(p
A
is a norm, then
P2
is the
Pl ^ P2
is the infimum of
A
if and only if
Pl"
p2
COROLLARY 5.13.
If
PROPOSITION 5.14.
{pj
infs(Pj)
Wo
PI’P2
If
A
In this sense,
and
c_C S
such that
0 < a _<
we have
P2
Pl
A
"
a
aioi-< Oi" Pi
and
aPi-< Oi" Oi
P2
However, by 2.8 we have that
W
^ P2 e Po
pl
then
S
in
pj
Wo
e
Pl ^ P2
then the infimum of the
In particular,
S
is equivalent
such that
ai-<
Pl ^ P2 is equivalent to PI" ^ P2"
Hence,
(Pl ^ P2
Pl ^ P2
Pl ^ P2
(recall 5 .6)
k
norm, then
is a
IA2
i.e.,
notationally,
%
0 < a
exists
there exists
a
then
belongs to
[5,p.472], i.e., there
e W
and
is closed under the formation of finite infima.
Recall that
pl,O2
Io
Ro
in
pl,P2
Thus,
Ro
Ol,p2
i
;
p
o
PROPOSITION 5.12.
Thus, for
then by 5.2
R
if
Q.
3
"
{pj}
is a lower bound for
infi(oj
infimum of the
to
p
and
COROLLARY 5.10.
pl,O2
n
is
infs(PI’P2)
Pl ^ P2
is closed under the formation of arbitrary
infima.
PROOF.
We have seen that
any lower bound for
Consequently,
p"
{pj}
(sup(p))
PROPOSITION 5.15.
S
O
is also in
P
then
O
is a lower bound for
p’
i e
>_
pj
j
J
If
{pj
so that
p,
>_
p
in
S
sup(p
p
The infimu of a family in
S
o
which is bounded below
This is not true for arbitrary families in
S
O
(5.8),
is
443
LATTICE THEORY OF FUNCTION SEMI-NORMS
bounded below in
sup(oj’)
that
such that
’
where
o
By 5.14,
PROOF.
S
p-< @j
j s J
A
Thus,
it is necessary and sufficient
o
j s J
So
sup(@)
@j
Therefore,
sup(o)
in
’
so that
is also saturated
If
@l,P2
s A
then
Pl
A
is the infimum of
P2
@i,2
is closed under the formation of finite infima.
Are
QUESTION 5.17.
6.
and
is a norm.
PROPOSITION 5.16.
A
{pj} _c So
for
By hypothesis, there exists
p’
Hence,
[5,p.458].
is saturated
s S
[5,p.458].
be saturated
S
belongs to
In order that
o
[5,p.454], i.e.,
in
infs(Pj
P ,R
o
closed under the formation of finite infima?
SUBLATTICES AND COMPLETENESS
We are now prepared to determine (as well as we can) which of the subsets
introduced in Section 3 are sublattices of
For those which are, we are also
P
interested in whether or not they are complete.
This section is essentially a
lattice-theoretic summary of the previous three sections.
A
With regard to completeness, it is worthwhile recalling the following.
lattice is complete if it is closed under arbitrary suprema and infima.
A
lattice is Dedekind complete if it is closed under arbitrary suprema and infima
of collections which are suitably bounded above for suprema and below for infima.
Since
same
s A
s A
and
o
these two notions of completeness are usually the
but not always.
As a consequence of the results of Section 5, we see that it is possible
for a subset of a sublattice to have an infimum in the sublattice which is
Therefore, it is possible for
different from its infimum in the whole lattice.
a sublattice to be a complete lattice, but not a complete sublattice!
P
THEOREM 6.1.
infimum given by
T
REMARKS 6.2.
do not know if
P
is a complete lattice with supremum given pointwise and
of Section 5.
(See 2.5, 2.6, 4.1, and 5.7.)
It is not clear whether
o
P
o
is a sublattice of
Dedekind complete in the sense that:
since we
For this reason, it is also
is closed under finite infima.
difficult to determine the sublattice status of
P
R ,W
o
So,Ao
However,
P
o
(i) it is closed under the formation of
is
I.E. SCHOCHETMAN and S. K. TSUI
444
arbitrary pointwise infima and (ii) it is closed under the formation of
P
y-infima of arbitrary collections bounded below in
I
THEOREM 6.3.
(4.2, 5.9), but
of Section 5.
k
infimum given by
Let
X
Y
supf(n)
S
all
is also a complete lattice
P
+
M
f
For the collection
j
(next example).
j
1,2
{pj
j
lq}
the corresponding
is given by
We may verify that
W
THEOREM 6.5.
in general
I
3"
i.e.,
is a sublattice of
is a sublattice of
S
o
infimum given by
of Section 5.
See 4.4 and 5.15.
PROOF.
which is not (Dedekind) complete
W
with supremum given pointwise and
k(f)
s S
all
o
A
is also a complete lattice, but not
S
is not complete in
{On
W [4,p.150], k
o # k
W
can be seen from
we may verify that
+
sup(f(n)) + lim sup(f(n))
k
have another example of
THEOREM 6.7.
That
%,(f)
n
S
Thus,
For this choice of
the example in 7.3.
PROOF.
I
W
a complete sublattice of
n
I
infp(@j)
(4.3, 5.12).
THEOREM 6.6.
Then
+
f s M
lim sup(f(n))
%’(f)
7.
I
Thus,
and define
lq
oj(f)
pj
o
with supremum given pointwise and
not a complete sublattice of
EXAMPLE 6.4.
so that
P
is a sublattice of
S
while
f s M
o
S
Thus we
(recall 5.5).
is a sublattice of
S
which is not (Dedekind) complete.
See 4.6 and 5.16.
DEMORGAN’S LAWS
In view of 2.8, it is natural to ask if
Once again, let
{j
j
J}
c__
p
DeMorgan’s Laws
The next theorem shows in what sense one
of the two DeMorgan Laws is valid.
THEOREM 7.1.
In general,
hold in general.
[infp(@j)]’
sup(p’)j
LATTICE THEORY OF FUNCTION SEMI-NORMS
PROOF.
For convenience, let
y
infp(pj)
445
sup(p)
and
We then
have
y’ (f)
To show
%’’ >-
Oj’
%"
j e J
Suppose next that
Zpj(gj)
%’’(f)
Jv
IfgldL
%’’
first observe that
Igl
J-decomposition of
assume
sup
<
also.
f
+
M
in
(f)
If
]fgld
X
%" _<
since
+
M
%’(g) <_
@j’(f)
If
g
+
M
@j
i.e.,
and
(gj)
(f)
j e J
<
f
is a
%’(g) _<
Since
j s J
<
then
<
>_
of finite type.
pj(gj)
i.e.,
f s
Y(g) -< i}
we may
then
and hence
Zfj fgjd
(finite sum)
Zj pj’(f) @j(gj)
(Hider’ s Inequality)
X
_<
-< (f)
Z
J
oj(gj)
Thus, from the definition of
%"
it follows that
Ifgldg -<
(f)%’(g)
_<
y’(f)
-< (f)
(f)
i.e.,
This theorem yields an alternate proof of the equality in 5.3.
Specifically,
we have
COROLLARY 7.2.
PROOF.
that
%’"
Let the notation be as in 5.3.
Since
%"
(sup(pj))
infp(pj)
we have
%’’
Then
sup(oj’)
%,"
o
by the theorem, so
o
In general, as the next example verifies, the other DeMorgan Law is false.
EXAMPLE 7.3.
Let
@n(f)
X
IN
Define
sup(f(k)) + sup(f(k))
k
k> n
f s
M’
n
1,2
I.E. SCHOCHETMAN and S. K. TSUI
446
We may verify that:
each
(i)
is an SFP norm, i.e.,
Pn
S
On
o
(k)) + lim sup(f (k))
infp(pn )(f) sup(f
k
k
infp(pn does not have the SFP.
(ii)
(iii)
o
n
Consequently, letting
n)
1,2
f
M
+
infp(@n)
infp(On
can’ t
Therefore, (sup(On))
does not
we see that
On
have the SFP while (sup(o
n
does.
equal
infp (On’)
Despite the previous example, there is a partial result of interest.
{Oj J
infs(Oj)’ sup(oj)
LEMMA 7.4.
However,
If
J}
!
P
-<
infs(Oj’)
(apply 2 .7).
(sup(oj))’
it is true that
{0j}
Of course, for finite families
(sup(0j))’
then
infp(Oj)
(recall 2.8).
infs(@j)
CONCLUDING REMARKS.
8.
A lattice-theoretic question which we have not yet considered is whether
As indicated by Luxemburg in [3],
(or any of its sublattices) is distributive.
the answer is
in
Ao
"no
More specifically, there exist norms
in general.
P
01,02,03
for which
(Pl
v p2
A
P3
(Pl
>
A
p3
V
(P2
A
p3
As a consequence of this, the
thus violating one of the distributive laws.
other distributive law is also violated, in general, because
(Pl ^p)
v
P3
<
01
Therefore, both distributive laws fail in
v
p) ^ (’P2
A
v
P3’)
and hence, in
o
P
and each of
its sublattices of interest.
Finally, observe that at first glance, finite sums seem to have nothing to
do with lattice theory of function semi-norms.
true
If
{01
sup(pl,...,pn
infp(@ I,... ,)
,@n
c p
then
01
+
+
Consequently, its associate
However, quite the opposite is
On e P
(@i
and is equivalent to
+
+
On)’
is equivalent to
In view of the previous, it is of interest to know which of the subsets of
447
LATTICE THEORY OF FUNCTION SEMI-NORMS
(introduced in Section 2) are closed under equivalence.
P
P
O
R, I, W are closed under equivalence;
ACKNOWLEDGEMENT.
S
isn’t.
Is
The spaces
A
Both authors were partially supported by Oakland University
Research Fellowships.
The authors are grateful to W.A.J. Luxemburg for providing them with a
copy of
[3],
as well as for a very helpful personal communication.
REFERENCES
I.
Halmos, P.R., Measure Theory
2.
Kothe, G., Topological Vector Spaces I, Springer-Verlag, New York, 1969.
3.
Luxemburg, W.A.J., Sup and
Van Nostrand, Princeton, 1962.
inf
constructions for saturated function norms,
unpublished preprint.
4.
Luxemburg, W.A.J., and Zaanen, A.C., Notes on Banach function spaces I, II,
Indag. Math. 25(1963), 135-153.
5.
Zaanen, A.C., In_tegration, North Hollard, Amsterdam, 1967.