Page 103 - Textos de Matemática Vol. 34

ORTHOGONAL POLYNOMIALS 93
vector space all the norms are equivalent, we adopt (without loss of generality)
n (2.1) ∥f∥n := |aν|
ν=0
foranarbitrarypolynomialf ∈Pn suchthatf(x)≡nν=0aνxν. Then(Pn,∥·∥n) is a Banach space (remember that a finite dimensional normed space is always complete). It is clear that Pn ⊂ Pn+1 for all n = 0, 1, 2, · · · (strict inclusion) and the topology of each Pn is identical to the one induced by Pn+1. Further, each Pn is closed in Pn+1. Therefore, since
+∞
P = Pn ,
n=0
the theory of l.c.s.’s leads us to consider in P its natural topology of the hyper-strict inductive limit1 defined by the sequence {(Pn , ∥ · ∥n )}n∈N0 (cf. Maroni [13],[15]; Treves [21, pp.126-131]). Since all the spaces Pn are Fr´echet (because they are Banach) one sees that P is an LF space (i.e., a strict inductive limit of a sequence of Fr´echet spaces), and so it is a complete space (see [21, p. 129] or [18, p. 146]).
Further, P is not metrizable (cf., e.g., [19, p. 29]).
Let P∗ be the algebraic dual of P, i.e., the set of all linear functionals u : P → C.
Given an u ∈ P∗, the action of u over a polynomial f will be denoted by ⟨u,f⟩. In particular, for each n ∈ N0,
un :=⟨u,xn⟩
will be called the moment of order n of u. The topological dual of P will be represented by P′ and the topology to be considered in this space is the dual weak topology, which, by definition, is characterized by the family of seminorms ℘:={pf :f∈P},where
pf(u):=|⟨u,f⟩|, u∈P′, f ∈P
(see Treves [21, p.197]). Alternatively, setting I := { | · |n : n ∈ N0 }, where
|u|n := sup |⟨u,xν⟩|, u ∈ P′ , n = 0,1,2,··· , 0≤ν ≤n
the following proposition holds (cf. Maroni [15]).
Theorem 2.1. The families of seminorms ℘ and I on P′ are equivalent. As a consequence P′ is a Fr´echet space and a metric can be explicitly defined by
+∞ 1 |u−v|n ′ ρ(u,v):= , u,v∈P .
n=0 2n 1+|u−v|n
The term “hyper-strict” is used for the strict inductive limit X = ind limnXn of a family of l.c.s.’s (Xn)n when each Xn is closed in Xn+1.