Summary: 1. Differentiation of vector valued functions of a real variable.
Definition 1.1. Suppose A R, E is a normed vector space,
f : A E.
We let
f = (a, m) : a int A and m = lim
xa
f(x) - f(a)
x - a
.
Note that f is a function. We say f is differentiable at a if a is in the domain
of f . For each nonegative integer m we define f(m)
by setting f(0)
= f, f(1)
= f
and requiring that f(m+1)
= (f(m)
) .
Theorem 1.1. Suppose A R, E is a normed vector space, f : A E and f is
differentiable at a. Then
lim

Source: Allard, William K. - Department of Mathematics, Duke University

Collections: Mathematics