Page 106 - Textos de Matemática Vol. 34

96 JOSE´ CARLOS PETRONILHO
δn,ν being the usual Kronecker symbol.
Theorem 3.1. Let {Rn}n≥0 be a simple set in P and {an}n≥0 be the associated dual basis in P∗. Then for every v ∈ P∗ the representation
∞
(3.1) v = λn an , λn := ⟨v,Rn⟩ (n = 0,1,2,···) n=0
holds, in the sense of the weak dual topology in P′.
Proof. We first notice that, according to (2.3), the statement in the theorem
makes sense. To prove (3.1) we have to show that
N
lim v−λνaν =0 in P′.
N →+∞
ν=0
Taking into account the definition of convergence in the dual weak topology in P′
and using the fact that {Rn}n≥0 is a basis in P, this is equivalent to show that N
(3.2) We have
hence
lim ⟨v−λνaν,Rn⟩=0, n=0,1,2,···.
proves (3.2).
ν=0
4. Some basic operations in P and P′

N →+∞
N ⟨v−
ν=0
ν=0
λν aν,Rn⟩= N
⟨v,Rn⟩ if n>N
⟨v,Rn⟩−λn if n≤N,
lim ⟨v−λνaν,Rn⟩=⟨v,Rn⟩−λn, n=0,1,2,···.
N →+∞
Therefore, (3.2) holds if and only if ⟨v,Rn⟩ = λn for all n = 0,1,2,···, which
As usual, X and Y being t.v.s.’s, the space of all bounded linear transformations of X into Y will be denoted by B(X,Y). We are particularly interested in the following transformations in B(P, P):
f→gf, f→Df, f→θcf,
where g is any fixed polynomial, c is any fixed complex number, and
gf(x) ≡ (gf)(x) := g(x)f(x) , Df(x) := f′(x) , θcf(x) := f(x) − f(c) . x−c
By duality the following (dual) transformations belongs to B(P′,P′): u→gu:=u◦g, u→Du:=−u◦D, u→(x−c)−1u:=u◦θc .