Page 105 - Textos de Matemática Vol. 34

ORTHOGONAL POLYNOMIALS 95
The diagram below shows the “right place” where the spaces P and P′ belong, showing also some relations between the topological spaces that we have referred.
metric
t.v.s.
l.c.s.

LF P
normed

Fr´echet P ′
Banach
3. Dual basis
The degree of any polynomial f ∈ P will be denoted by ∂f. Any sequence of polynomials {fn}n≥0 such that ∂fn = n for all n will be called a simple set. Let {Rn}n≥0 be a simple set of polynomials. Since it is an (algebraic) basis for P we can consider the corresponding dual basis in P∗, say {an}n≥0, where, by definition, an : P → C is the linear functional characterized by
⟨an,Rν⟩ := δn,ν (n,ν = 0,1,2,···),
96
discret
87