By Cora Sadosky

ISBN-10: 0824783026

ISBN-13: 9780824783020

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**Sample text**

2) , sketch τ DH and compare this to τ DH . 12 (Continuation on Reinhardt domains). Let D ⊂ Cn be a Reinhardt domain containing the origin and D its polybalanced hull. Then the following holds: 1. For all f ∈ O (D) the series α∈Nn Dα f (0) α z α! converges to f compactly on D. 2. The restriction → O (D) , F → F |D ρ:O D is an isomorphism of algebras. Proof. 1. Since 0 ∈ D there is a polydisc P ⊂ D containing 0 such that the Taylor expansion Dν f (0) ν f (z) = z ν! n ν∈N converges compactly on P.

Since 0 ∈ D there is a polydisc P ⊂ D containing 0 such that the Taylor expansion Dν f (0) ν f (z) = z ν! n ν∈N converges compactly on P. Thus, for all α ∈ Zn and all z ∈ P we have fα (z) = = 1 2πi n 1 2πi n Tn f (ζz) dζ ζ α+1 dζ Tn ζ α+1 ν∈Nn converges uniformly for ζ∈Tn n ν ν = ν∈Nn Dν f (0) ν ν ζ z ν! D f (0) ν 1 z ν! 4 = D α f (0)z α , α! 0, if α ∈ Nn . if α ∈ / Nn Since D is a domain and P ⊂ D is open this equality holds on all of D. 27 implies f (z) = fα (z) = α∈Zn fα (z) = α∈Nn α∈Nn Dα f (0) α z α!

Xp ∈ A such that p K⊂ Bδ (xj ). j=1 =:Kj (fj )j≥0 is pointwise a Cauchy sequence on A, hence, there is N = Nε ∈ N such that ε fk (xj ) − fl (xj ) < 3 for all k, l ≥ N and all j = 1, . . , p. If x ∈ K there is an index q ∈ {1, . . , p} such that x ∈ Kq , which implies d (x, xq ) ≤ δ. Now we can conclude that for all k, l ≥ N and all x ∈ K, fk (x) − fl (x) ≤ fk (x) − fk (xq ) + fk (xq ) − fl (xq ) + fl (xq ) − fl (x) ε ε ε < + + = ε. 27 (Arzel`a–Ascoli). , there is a constant C > 0 such that sup sup fj (x) ≤ C.

### Analysis and partial differential equations. Dedicated to Mischa Cotlar by Cora Sadosky

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