In this paper we treat that portion of classical invariant theory which goes under the name of "first" and "second" fundamental theorem for the classical groups, in a characteristic free way, i.e., where the base ring A is any commutative ring (in particular the integers 77 or an arbitrary field). The results we obtain are exactly the ones predicted by the classical theory (see [5]), provided we interpret the word invariant to mean formal or absolute ones, they are contained in Theorems 3.1, 3.3, 4.1, 5.6, 6.6. For instance we have THEOREM 3.1 [5]. The ring of polynomial functions over A in the entries of n m-vectors Xl ,..., x n and n m-covectors ~1,..., ~ , left formally invariant under the action of GL(m, -) is generated over A by the scalar products
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