Appendices

Abstract

We briefly recall here the two key theorems about continuous functions used in the book. Recall that a subset of a topological space is relatively compact (resp. relatively sequentially compact) if its closure is compact (resp. sequentially compact). Also, compactness and sequential compactness are equivalent notions for metric spaces. The first theorem concerns the relative compactness in the space C([0, T];ℛ^d) of continuous functions [0, T] → ℛ^d for a prescribed prescribed T > 0. C([0, T];ℛ^d) is a Banach space under the ‘sup-norm’ $\left\| f \right\|\mathop = \limits^{def} {\sup _{t \in \left[ {0,T} \right]}}\left\| {f\left( t \right)} \right\|$. That is,

• (i) it is a vector space over the reals,

• (ii) ∥ · ∥ :C([0, T];ℛ^d) → [0, ∞) satisfies

  • (a) ∥f∥ ≥ 0, with equality if and only if f ≡ 0,

  • (b) ∥αf∥ = |α|∥f∥ for α ∈ ℛ,

  • (c) ∥f+g∥≤ ∥f∥+∥g∥,

  and

• (iii) ∥ · ∥ is complete, i.e., {f _k } ⊂ C([0, T];ℛ^d), ∥f _m − f _n ∥ → 0 as m, n → ∞, imply that there exists an f ∈ C([0, T];ℛ^d) such that ∥f _n − f∥ → 0 (the uniqueness of this f is obvious).