A sequence of modals, [??] = ([[??].sub.k]) [member of] w(gI), is said to be modal fundamental sequence if for every [epsilon] > 0 there exists [k.sub.0] [member of] N such that d([[??].sup.k], [[??].sub.n]) < [epsilon] whenever n, k > [k.sub.0].

Let {[[??].sup.(i)]} be any fundamental sequence in the space h(gI), where {[[??].sup.(i)]} = {[[??].sup.(i)], [[??].sup.(i).sub.1], ...}.

which leads to the fact that {[(T[[??].sup.(i)]).sub.k]} is a fundamental sequence in gI for every fixed k [member of] N.

We also call fundamental sequence of X the integer sequence

We say that X is parallel unimodal if the fundamental sequence of X is a unimodal sequence; we also say that X is parallel symmetric if the fundamental sequence of X is a symmetric sequence.

It follows that the parallel rank of P(n, r) is equal to the maximum of the length of the two sequences (13) and (14) and that the fundamental sequence is the sum of the two sequences.

The case of Frechet-Schwartz spaces follows analogously if, instead of a fundamental sequence of bounded sets, we take a fundamental sequence of zero neighbourhoods.

By the discussion at the end of Section 2 X[[??].sub.[pi] Y is an LS-space and by [15,41.4(7)] [GAMMA]([A.sub.n] [cross product] [B.sub.n]) is a fundamental sequence of bounded sets (here [GAMMA] stands for the absolutely convex hull).

The polynomials also may be defined by an orthogonal ization procedure, if it is applied to the fundamental sequence {[x.sup.k]} in the order of power increase.

In particular, the algorithm of inverse orthogonalization of the fundamental sequence results in redistribution of the zeros.

Let [([U.sub.n]).sub.n [member of] N] be a

fundamental sequence of 0-neighborhoods in E.