Now, we are not assuming access to bases for vector spaces formula_8 and formula_9 that we want to form the tensor product formula_22 of. Instead, we will take "all" of formula_8 and formula_9 as "basis" to build up the tensors. This is the next best thing and the one thing we are "guaranteed" to be able to do, regardless of any concerns in finding a specific basis; this corresponds to adding together arbitrary outer products formula_19 of arbitrary vectors in the last part of the "Intuitive motivation" section. The only difference here is that if we use the free vector space construction and form the obvious formula_61, it will have many redundant versions of what should be the same tensor; going back to our basisful case if we consider the example where formula_62 in the standard basis, we may consider that the tensor formed by the vectors formula_63 and formula_64, i.e.