Almost Periodic Differential Equations by A.M. Fink

By A.M. Fink

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Since all norms in Cm are equivalent it suffices to consider the maximum norm . 4. O (U ) as a topological space 29 and f |Pr (a) ∞ ≤M for all f ∈ F. This is possible, because U is open, Pr (a) is compact and F is locally bounded. Put K := P r2 (a) and let ε > 0, f ∈ F and x, y ∈ K be arbitrary. Applying the Mean Value Theorem in R2n to f gives f (x) − f (y) ∞ ≤ x−y ∞ sup Df (z) , z∈K where sup Df (z) = sup z∈K sup z∈K x Df (z) x ∞ =1 ∞ . Assume without loss of generality that n Df (z) x = k=1 ∂f1 (z) xk , .

N} . 10. Show that the polybalanced hull of a Reinhardt domain is again a Reinhardt domain. 1. 11. 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!

Let U = ∅. , there is some j1 ∈ N+ , such that 1 dist . ∞ (z, Cn \ U ) ≥ . j1 Also, there is some j2 ∈ N+ ,such that z ∞ ≤ j2 . Then z ∈ Kj3 , where j3 := max {j1 , j2 } . Since every Kj is a subset of U, 3. follows. 4. If K is a compact subset of U the sets Kj◦ j ∈ N+ form an open cover of K. Compactness of K implies the existence of some jK ∈ N+ , such that jK K⊂ Kj , j=1 but since the Kj ⊂ Kj+1 for all j this implies K ⊂ KjK . 15. A sequence of compact sets having the properties 2. and 3. 14 is called a compact exhaustion.

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