An Example of a Bounded Set Which is Not Totally Bounded
We write the set of infinite sequences as setr^infinitysuch that
We can define a metric
The metric spaceis called the Hilbert space
Letbe the subset ofconsisting of the elements
andis bounded because the diameter ofand A is bounded.
Takethen the- net ofconsists of all elements ofThe infinite setcannot be separated into a finite number of subsets, each with diameter less thansois not totally bounded.