7.3 The Holder Spaces
We consider these spaces as spaces of functions defined on an interval
although one could have
just as easily. A slightly more general version is
in the exercises. They are a very interesting example of spaces which are not
Definition 7.3.1 Let p > 1. Then f ∈ C1∕p
means that f ∈ C
Then the norm is defined as
We leave it as an exercise to verify that C1∕p
is a complete normed linear
Let p > 1. Then C1∕p
is not separable. Define uncountably many functions, one
is a sequence of −
1 and 1.
Thus εk ∈
if the two
sequences differ in at least one slot, one giving 1 and the other equaling −
Then this is 1∕p Holder. Let s < t.
If t = 1 and s = 0, there is really nothing to show because then the difference equals 0.
There is also nothing to show if t = s. From now on, 0 < t−s < 1. Let k0 be the largest
integer which is less than or equal to
. Note that
0 because 0 < t − s <
Now k0 ≤−log 2
+ 1 and so −k0 ≥
Using this in the sums,
is indeed 1∕p
Now consider ε≠ε′. Suppose the first discrepancy in the two sequences occurs with εj.
Thus one is 1 and the other is −1. Let t =
Now consider what happens for k > j
for some integer m. Thus the whole mess reduces to
which shows that
Thus there exists a set of uncountably many functions in C1∕p
and for any two of
is not separable.