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A Chu space over a set  is a triple
 with
 .
 is called the carrier and
 the cocarrier.
Although the definition is symmetrical, in practice asymmetric uses are common. In particular, often
 is just taken to be a set of function from
 to  , with
 (such a Chu space is called normal and is abbreviated
 ).
We define the perp of a Chu space
 to be
 where
 .
Define  and  to be functions defining the rows and columns of
 respectively, so that
 and
 are given by
 . Clearly the rows of
 are the columns of
 .
Using these definitions, a Chu space can be represented using a matrix.
If  is injective then we call
 separable and if  is injective we call
 extensional. A Chu space which is both separable and extensional is biextensional.
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