A Chu space resembles a formal language, in that it may be understood
intuitively as a set of ``words'' over an alphabet .
An ordinary word *w* of length *n* can be defined as a function
,
having for its *i*-th symbol *w*(*i*)
or *w*_{i}.

Chu spaces modify this in two ways. First, a word is taken to be a
function
where *X* is an arbitrary set, not necessarily
an initial segment of the positive integers. We may then speak of *X*
as the ``length'' of *w*. (Note that the number of words of ``length''
a five-element set is the same as the number of ordinary words of ordinary
length 5.)

Second, the words of a given Chu space are all of the same length,
i.e. all have a given set *X* as their common domain. No restrictions
are placed on either *X* or ,
which may be empty, or initial
segments of the positive integers, or sets of reals, or any other set.
Likewise no restrictions are placed on the words, which may be any
function from *X* to .

This intuition is formalized as follows. A *Chu space over*
is a triple
consisting of sets *A* and *X* and a function
.
We call *A* and *X* respectively the *carrier*
and *cocarrier* of ,
their elements respectively *points*
and *states*, and *r* the *interaction matrix*.

*Words and dual words.* The interaction matrix has *left*
and *right transposes*
and
satisfying
= *r*(*a*,*x*) =
,
which we may
interpret as representations of *A* and *X* respectively. For each point
,
represents *a* as a function from *X* to ,
i.e. of type ,
namely a word over an alphabet
of length *X* in
the above sense. Dually
represents state *x* as a function
from *A* to ,
similarly constituting a word of length *A* over the
same alphabet ,
which we shall refer to as a *dual* word of .

The Chu space whose words are the dual words of
is
where
,
called the *dual* of
and denoted
.

When two points *a*,*b* are represented by the same word, i.e. when
,
we call them *equivalent*, written ,
Dually, equivalent states, those satisfying
,
are likewise indicated by .

When no two distinct points are equivalent, i.e.
is injective, we call
a *faithful* representation of *A*,
and say that
is *separable*. Dually when
is
injective we call it a faithful representation of *X*, and say that
is *extensional*. A Chu space that is both separate and extensional
is called *biextensional*.

The usual notion of a formal language as a set of words all distinct corresponds to the property of separability. Nonseparable Chu spaces may be understood as multisets of words, allowing the same word to occur more than once in the language. For our purposes the identity of points and states is determined by their representations, and for this reason the biextensional Chu spaces will be the ones we shall be mainly concerned with.

The alphabet itself forms a language consisting of all words of length 1 over alphabet . This makes it a Chu space, which we denote where and is projection on the first coordinate, satisfying for all . Its dual, , consists of a single word containing one occurrence of every symbol in . Its role is as the discrete singleton, denoted by 1.

The discrete empty language, denoted 0, is . The inconsistent empty language, , plays no important role and needs no name.