Construction
Given a set, we can construct a free -module over . The module is simply the direct sum of copies of, often denoted . We give a concrete realization of this direct sum, denoted by, as follows:
- Carrier: contains the functions such that for cofinitely many (all but finitely many) .
- Addition: for two elements, we define by .
- Inverse: for, we define by .
- Scalar multiplication: for, we define by .
A basis for is given by the set where
(a variant of the Kronecker delta and a particular case of the indicator function, for the set ).
Define the mapping by . This mapping gives a bijection between and the basis vectors . We can thus identify these sets. Thus may be considered as a linearly independent basis for .
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