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.