189
Internat. J. Math. & Math. Sci.
(1985) 189-191
Vol. 8 No.
TWO LARGE SUBSETS OF A FUNCTIONAL SPACE
F.S. CATER
Department of athematics
Portland State University
Portland, Oregon 97207 U.S.A.
(Received May 29, 1984)
ABSTRACT.
Let
[0,I] such that the
[0,i].
vanishes is dense in
Let
D
D
unsigned on every interval, and let
with void interior.
D
O
Neither
Then
D
nor
I
D
O
and D
PI
that
G6-subsets
are dense
I
f
that are
consist of those functions in
I
of
is a subset of the other.
KEY WORDS AND PHRASES. Banach space of functions, derivative dense
1980 MATHEMATICS SUBJECT CLSS9FICATION: 26A24
i.
PI
consist of those functions in
O
vanish on dense subsets of measure zero.
PI
set of points where
of
f
denote the Banach space composed of all bounded derivatives
PI
everywhere differentiable functions on
G-.b.at.
INTRODUCTI ON.
functions on
of all bounded derivatives of everywhere differentiable
D
The real vector space
is a Banach space [i] under the norm
[0,i]
II f II
(At the endpoints
Salat
0
and
sup
If(x) l.
0_<x_<l
we require that the one sided derivatives exist.)
I
Tibor
essentially proved [I] that the set
D
{f
O
D:
D.
is a nowhere dense subset of
PI
{f
is unsigned on any interval}
f
D: f
To do this, he observed that
0 on a dense subset of
D
is a nowhere dense Banach subspace of
and
D
in its own right, it is natural to study
DI
{f
PI:
In this note, we prove that D
O
THEOREM i. D
is a dense
O
THEOREM 2. D
is a dense
I
O
DO PI"
D
G6-set
C6-set
Since
as a subset of
f # 0 almost everywhere on
and
[0,I]}
PI
PI"
is a Banach space
Put
[0,i]}.
are "large" subsets of the Banach space
I
in
in
PI with void interior.
PI with void interior.
PI"
F.S. CATER
190
If
E
{f
E
Now put
-
PI’ then
\
PI DI" We
f
PI:
is not almost everywhere discontinous}.
f
2, a result of Clifford Weil [2].
obtain from Theorem
COROLLARY i (C. Weil).
D
D
or
O
is a subset of the
I
PI"
is a first category subset of
E
Finally, we show that neither of the
Clifford Weil [3] proved most of Theorem i.
sets
is continuous, and
f
must vanish at every point where
f
spaces of nonnegative derivatives.
other, and we prove an analogue of Theorem 2 for
(I take this opportunity to thank the referee for
simplifying many of my arguments.)
[3],
[a,b]
easy to find an interval
g
[a,b].
Let
D
has void interior.
O
f(b)
for which f(a)
f(x)
g(x)
Now let
but
(D
g
[a,b],
x
if
I
09 DI).
PI
f
DI PI \)n En"
PI
(f
< e.
0
mA
k
E
Thus
n
n
-I
Then at each
e
c
and
el
PI"
n-l}.
PI"
is closed in
fk En
Let
and
O}
fk(x)
Ak, f(x)
x
A
E
is nowhere dense.
DI
g
to get a function
such that
Clearly
=Nj3k>
0.
But
mA
n
E
and
e
-i
so
f
En.
PI"
is closed in
It remains to prove that
[4]
[a,b].
Say
0.
{x:
and
En
for
e
x
if
It is
e > 0.
Thus every open set in
O} >
m(x: f(x)
PI:
We claim that each
limk_=ll fk -f I
and
is essentially
and
If(x)
0 and
g(x)
and
0U
D
En
Then
PI
in
f-gll
Finally,
void interior in
D
D l, so D
3
I has
O
Proof of Theorem 2. For each positive integer n, define
PI
contains functions
f
G-set
is a dense
so we leave it.
It remains to prove that
x
DO
The proof that
Proof of Theorem i.
given in
n
0 < g < i.
such that
If(x)
m{x: 0
c}
<
-I
n
-cg(x)}
m{x: f(x)
Let
f
n
But there is a number
Use
O.
c > 0
It follows that
< n
-i
Finally,
0}
m[x: f(x) + cg(x)
and
f
+ cg { En
II
Moreover
n
llcgll <-
cg)-fll
(f +
In the proof of Theorem i we saw that
-I
c
e. Thus E
DI,
D 13 D I, and hence
O
n
is nowhere dense.
has void
[]
interior.
It follows from Theorems i and 2 that
D
O
I-
DI,
the set of all functions in
P1
that vanish on dense sets of measure zero and are unsigned in any interval, is a
dense
C6-subset
of
PI"
Next we show that the sets
DO
and D
I
are quite different.
Neither is a subset of the other.
THEOREM 3.
The Sets
D
O
\ D
I
DI\
and
D
O
are nonvoid.
We construct a sequence of intervals
-n
0
h(bn)
and h(a
2
(a ,bn) with mutually disjoint closures, such that b -a
n
n
n n
means
where
Let
for each n and .n(an,hn) is dense n [0 I]
PROOF.
Let
h
be a function in
D
O
hn h(an,bn
TWO LARGE SUBSETS OF A FUNCTION SPACE
for
characteristic function:
P
h
n
-< x
0
(II hn If)
I’ and the sequence
m{[0
Thus
h(x)
hn(X)
In
2
-n
h
DO\
f e
D
for all
0
(an,bn),
f(x)
and
l]\hJn(a
n- bn)}_
It follows that
PI"
n
n
(x).
(an,bn
Put
cannot be signed on any subinterval of an
f
[0,1]\Un(an,bn) hn(X)
x
I,
is bounded
f=
Now
<
191
Z (b -a
i
n
n
D
f
so
O.
Clearly for
O
But
0
n
I.
We use [4]
DI\
to obtain a function in
D
O
Now put
Then
D
2
PROOF.
P2"
+ cg
D2
{f e
P2:
f is nonnegative},
0 almost everywhere on
f
D
E
We conclude by showing that
G6-subset
D
2
g e D
and
2
I
of
with void interior.
P2
as in the proof of Theorem 2.
n
[0,I]}.
P2"
is a dense
2
Define
It remains to prove that
to construct
f
PI:
"large" subset of
THEOREM 4.
of
{f
is a complete metric space in its own right.
P2
is a
P2
D
0
such that
(f + cg)
f
ll
P2\hJn (En C P2
2
-< g -< i.
II
For any
cgll -<
c.
n
P2
is dense in
f
D
So
E
Then
2
P2
and any
is a dense
is a closed subset
P2
c >
We use [4J
0, we have
G-subset
of
has void interior follows from the same proof (for Theorem I) that
That D
2
has void interior, so we leave this point.
Compare Theorem 4 to the work in [5].
form a dense
functions on
G6-subset
e 2.
D
O
There it is shown that the singular functions
of the complete metric space of continuous nondecreasing
[0,i] under the sup
are taken, [5] suggests that
D
2
metric.
is a
When the primitives of the functions in
"small" subset of
P2"
P2
Of course the metric
used in [5] was different from the one used here.
RE FE REN CE S
i.
SALAT, TIBOR, On functions that are monotone on no interval, Amer. Math. Monthly
88 (1981) 754-755.
2.
WEIL, CLIFFORD, The space of bounded derivative
Real Analysis Exchange
(1977-8) 38-41.
WEIL, CLIFFORD, On nowhere monotone functions, Proc. Amer. Math. Society 56
3.
,
(1976) 388-398.
4.
ZAHORSKI, Z., Sur la premiere derive,
Vol 69 (1950), 26, Lemma 11.
5.
ZAMFIRESCU, TUDOR, Most
88 (1981) 47-49.
Transactions of the
Amer. Math. Society,
monotone functions are singular, Amer. Math. Monthly