  We bound the number of electromagnetic signals which may be observed over a
frequency range $2W$ for a time $T$ within a region of space enclosed by a
radius $R$. Our result implies that broadband fields in space cannot be
arbitrarily complex: there is a finite amount of information which may be
extracted from a region of space via electromagnetic radiation.
  Three-dimensional space allows a trade-off between large carrier frequency
and bandwidth. We demonstrate applications in super-resolution and broadband
communication.
