  A fundamental problem in any communication system is: given a communication
channel between a transmitter and a receiver, how many "independent" signals
can be exchanged between them? Arbitrary communication channels that can be
described by linear compact channel operators mapping between normed spaces are
examined in this paper. The (well-known) notions of degrees of freedom at level
$\epsilon$ and essential dimension of such channels are developed in this
general setting. We argue that the degrees of freedom at level $\epsilon$ and
the essential dimension fundamentally limit the number of independent signals
that can be exchanged between the transmitter and the receiver. We also
generalise the concept of singular values of compact operators to be applicable
to compact operators defined on arbitrary normed spaces which do not
necessarily carry a Hilbert space structure. We show how these generalised
singular values can be used to calculate the degrees of freedom at level
$\epsilon$ and the essential dimension of compact operators that describe
communication channels. We describe physically realistic channels that require
such general channel models.
