The two Hilbert spaces H 0 {H_0} and H 1 {H_1} are defined to be a generating pair if H 1 {H_1} is a dense subspace of H 0 {H_0} and if the norm of an element in H 1 {H_1} is greater than or equal to the norm in H 0 {H_0} . It is shown that the pair generates a sequence of spaces { H k } , − ∞ > k > ∞ \{ {H_k}\}, - \infty > k > \infty , such that any two spaces of the sequence form again a generating pair. Such a pair is shown to generate, in turn, a subsequence of { H k } \{ {H_k}\} . Also, representation theorems are derived for bounded linear functionals over the spaces of the sequence { H k } \{ {H_k}\} , generalizing the Lax representation theorem and the Lax-Milgram theorem.