The Computable Differential Equation Lecture ... - Bruce E. Shapiro

CHAPTER 2. SUCCESSIVE APPROXIMATIONS 35
Distributivity For all a, b ∈ R and for all u, v ∈ V,
(a + b)v = av + bv (2.45)
a(u + v) = au + av (2.46)
Identity for Scalar Multiplication For all vectors v ∈ V,
1v = v (2.47)
Example 2.3. The usual Cartesian vector space to which we are accustomed is a
vector space with vectors being defined as ordered triples of coordinates 〈x, y, z〉.
Example 2.4. Show that the set F[a, b] of all integrable functions f : [a, b] ↦→ R is
a vector space.
Solution. Let f, g, h ∈ F[a, b] and c, d ∈ R Then
• V is closed: Let p(t) = f(t) + g(t) and q(t) = ch(t). Then p, q : [a, b] ↦→ R
hence p, q ∈ F[a, b]
• f(t) + g(t) = g(t) + f(t) so commutivity holds.
• (f(t) + g(t)) + h(t) = f(t) + (g(t) + h(t)) and c(df(t)) = (cd)f(t) so both
associative properties hold.
• The function f(t) = 0 is an additive identity.
• For any function f(t) the function −f(t) is an additive inverse.
• (c+d)f(t) = cf(t)+df(t) and c(f(t)+g(t)) = cf(t)+cg(t) so both distributive
properties hold.
• The number 1 acts as an identity for scalar multiplication.
Hence the set F[a, b] is a vector space.
Definition 2.3 (Norm). A norm ‖ · ‖ : V ↦→ R on a vector space V is a function
mapping the vector space to the real numbers such that
1. For all v ∈ V, ‖v‖ ≥ 0.
2. ‖v‖ = 0 if and only if v = 0.
3. For al v ∈ V and for all a ∈ R, ‖av‖ = |a| ‖v‖.
4. The norm satisfies a triangle inequality: for all v, w ∈ V,
‖v + w‖ ≤ ‖v‖ + ‖w‖ (2.48)
Definition 2.4 (Normed Vector Space). A vector space on which a norm has been
defined is a normed vector space.
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge