• D={z : ∣z∣<1}
• H (A→B) denotes the set of holomorfic
functions from A to B
• We define:
W={f∈H (D→R) : ∣∣f∣∣_W <∞ }
where ∣∣ ∙ ∣∣_W : { (W,→,R_+ ),(f, ,(Σ_(n=0) ^∞ ((∣f^((n)) (0)∣)/(n!)))) :}
Let f∈W
Show that ∀g∈H ( f(D^ )), g○f∈W
tip: show that
∣∣h∣∣_W ≤cste × Sup_(z∈D) {∣h(z)∣+∣h′′(z)∣}
and that W is an algebra
then, re−wright f=f_1 +f_2 with
f_2 : z Σ_(n=N) ^∞ ((f^((n)) (0))/(n!))z^n
with N great enough to make sure that
Σ_(n=0) ^∞ ((g^((n)) (0))/(n!))f_2 ^( n) is well defined and converges
over W.
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