By Péter Érdi

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**Additional resources for Complex Dynamics - Advanced System Dynamics in Complex Variables**

**Example text**

1/2 Let (H, · ) be an inner product space and set h = h, h . Then (H, · ) is a normed space. Let (H, ·, · ) be an inner product space and · the corresponding norm. Then we have 1. Polarization law : 2 2 4 h1 , h2 = h1 + h2 − h1 − h2 +i h1 + i h2 2. Parallelogram law : 2 2 2 2 2 h1 + 2 h2 = h1 + h2 − h1 − h2 . 2 −i h1 − i h2 2 , and Let (H, · ) be a normed space and define d(h1 , h2 ) = h1 − h2 . Then (H, d) is a metric space. Let (H, · ) be a normed space. If the corresponding metric d is complete, we say (H, · ) is a Banach space.

A closed path is a path whose beginning point is equal to its end point. If f is a C 0 −function on an open set U ⊂ MC admitting a holomorphic primitive g, and γ is any closed path in U , then γ f = 0. Let γ, η be two paths defined over the same interval [a, b] in an open set U ⊂ MC . Recall (see Introduction) that γ is homotopic to η if there exists a C 0 −function ψ : [a, b] × [c, d] → U defined on a rectangle [a, b] × [c, d] ⊂ U , such that ψ(t, c) = γ(t) and ψ(t, d) = η(t) for all t ∈ [a, b].

1/2 Let (H, · ) be an inner product space and set h = h, h . Then (H, · ) is a normed space. Let (H, ·, · ) be an inner product space and · the corresponding norm. Then we have 1. Polarization law : 2 2 4 h1 , h2 = h1 + h2 − h1 − h2 +i h1 + i h2 2. Parallelogram law : 2 2 2 2 2 h1 + 2 h2 = h1 + h2 − h1 − h2 . 2 −i h1 − i h2 2 , and Let (H, · ) be a normed space and define d(h1 , h2 ) = h1 − h2 . Then (H, d) is a metric space. Let (H, · ) be a normed space. If the corresponding metric d is complete, we say (H, · ) is a Banach space.