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];ℛd) of continuous functions [0, T] → ℛd for a prescribed prescribed T > 0. C([0, T];ℛd) 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];ℛd) → [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];ℛd), ∥f m − f n ∥ → 0 as m, n → ∞, imply that there exists an f ∈ C([0, T];ℛd) such that ∥f n − f∥ → 0 (the uniqueness of this f is obvious).
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.