Notes on Stability Analysis
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چکیده
1.1. Norms on Vector Spaces. We want to measure the “size” or “length” of a vector. The mathematical device for doing this in called a norm. Here is the definition. Definition 1.1. Let V be a vector space over K. A norm on V is a function ‖·‖ : V → R : v 7→ ‖v‖ that has the following properties. (1) ‖v‖ ≥ 0 for all v ∈ V , and ‖v‖ = 0 if and only if v = 0. (2) If λ ∈ K and v ∈ V , ‖λv‖ = |λ| ‖v‖. (3) For all v1, v2 ∈ V , ‖v1 + v2‖ ≤ ‖v1‖ + ‖v2‖. This is called the triangle inequality. Exercise 1.2. Show that ‖−v‖ = ‖v‖ and ‖v1 − v2‖ = ‖v2 − v1‖. The following is a simple consequence of the triangle inequality, which we will include under that name. Proposition 1.3. For all v1, v2 ∈ V , ∣∣ ‖v1‖ − ‖v2‖ ∣∣ ≤ ‖v1 ± v2‖. Proof. We have ‖v1‖ = ‖v1 − v2 + v2‖ ≤ ‖v1 − v2‖+ ‖v2‖, by the triangle inequality, so ‖v1‖ − ‖v2‖ ≤ ‖v1 − v2‖. Switching the roles of v1 and v2 gives ‖v2‖ − ‖v1‖ ≤ ‖v2 − v1‖ = ‖v1 − v2‖. Thus, we have ± [ ‖v1‖ − ‖v2‖ ] ≤ ‖v1 − v2‖,
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تاریخ انتشار 2003