Page 108 - Textos de Matemática Vol. 34

98 JOSE´ CARLOS PETRONILHO
The right-multiplication just defined as above enable us to introduce a product in P′. In order to see this, notice that for any fixed u ∈ P′ the linear transformation f → uf belongs to B(P, P). In fact, from (4.1) we easily deduce
∥uf∥n ≤(n+1)|u|n∥f∥n, u∈P′ , f ∈Pn (n=0,1,2,···);
hence, for every nonnegative integer number n the restriction to Pn of the mapping
f → uf is continuous. Henceforth, by duality we introduce the following Definition 4.3. Let u, v ∈ P′. The product uv is the functional in P′ defined by
⟨uv,f⟩ := ⟨u,vf ⟩ .
This product is commutative, since the moments of uv and vu coincide. Further, there exists an unit element in P′, namely the Dirac functional at the origin, δ ≡ δ0. As usual, for any c ∈ C, the Dirac functional at the point c, δc, is defined by
⟨δc,f⟩:=f(c), f∈P. Finally, we list the following properties
1. δf=f 6.
2. v(uf ) = (vu)f
3. (u+v)w=uw+vw
4. (uv)w=u(vw)
5. uhasinverseiffu0 ̸=0
f(uv) = (fu)v + x(uθ0f)v
7. D(uf)=(Du)f+uf′+uθ0f
8. D(uv)=(Du)v+uDv+x−1(uv) 9. (x−c)((x−c)−1u)=u
10. (x−c)−1((x−c)u)=u−u0δc , which hold for every f ∈ P, c ∈ C, and u,v,w ∈ P′ (see Maroni [14], e.g.).
5. The formal Stieltjes series
Denote by ∆′ ≡ C[[z]] the vector space of the formal series in the variable z with
coefficients in C,
∆′ :=ν≥0cνzν : cν ∈C forall ν=0,1,2,···.
In ∆′, the operations of addition, multiplication, and scalar multiplication are de- fined in the usual way (cf., e.g., Treves [21], Cartan [3]). Endowing ∆′ with the sequence of seminorms {sn : n ∈ N0 } , where
s c zν:= sup |c |, nν≥0ν ν
0≤ν ≤n
∆′ becomes a metrizable l.c.s. which can be identified with P′. In fact, the following proposition holds (cf. Maroni [11]).