We write the set of infinite sequences as setr^infinity
such that
Let
We can define a metric
The metric space
is called the Hilbert space
Let
be the subset of
consisting of the elements
Then
andis bounded because the diameter of
and A is bounded.
Take
then the
- net ofconsists of all elements of
The infinite setcannot be separated into a finite number of subsets, each with diameter less than
sois not totally bounded.