22.4. THE DERIVATIVE OF A COMPACT MAPPING 713

Dg(f(x))(f(x+v)− f(x))+o(f(x+v)− f(x))= Dg(f(x)) [Df(x)v+o(v)]+o(f(x+v)− f(x))= Dg(f(x))Df(x)v+o(v)+o(f(x+v)− f(x)) (22.3.11)= Dg(f(x))Df(x)v+o(v)

By Lemma 22.2.3. From the definition of the derivative D(g◦ f)(x) exists and equalsDg(f(x))Df(x).

22.4 The Derivative Of A Compact MappingHere is a little definition about compact mappings. It turns out that if you have a differen-tiable mapping which is also compact, then the derivative must also be compact.

Definition 22.4.1 Let C ∈ L (X ,Y ) . It is said to be compact if it takes bounded sets toprecompact sets. If f is a function defined on an open subset U of X , then f is calledcompact if f (bounded set) = (precompact) .

Theorem 22.4.2 Let f : U ⊆ X → Y where f takes bounded sets to precompact sets. ThenD f (x) also takes bounded sets in X to precompact sets in Y.

Proof: If this is not so, then there exists a bounded set B in X and for some ε > 0 asequence of points D f (x)bn such that all these points are further apart than ε . Without lossof generality, one can assume B = B(0,r) , a ball. In fact, one can assume that r > 0 isas small as desired because if D f (x)B(0,r) is precompact, then so is D f (x)B(0,R) ,R >

r. Just get an εrR net {D f (x)xn}N

n=1 for D f (x)B(0,r) and consider{R

r D f (x)xn}N

n=1 .

∪nB(D f (x)xn,ε

rR

)covers D f (x)B(0,R), so ∪nB

(Rr D f (x)xn,ε

)covers D f (x)B(0,R).

Choose r very small so that r < ε/4 and

f (x+ xn)− f (x) = D f (x)xn +o(xn) , ∥o(xn)∥< ∥xn∥

and there are infinitely many D f (x)xn further apart than ε,xn ∈ B(0,r). Then considerB(x,r) and { f (x+ xn)}∞

n=1 .

∥ f (x+ xn)− f (x+ xm)∥ ≥ ∥D f (x)xn−D f (x)xm∥−∥o(xn)−o(xm)∥

≥ ε−2ε

4=

ε

2

contradicting the assertion that f takes bounded sets to precompact sets.

22.5 The Matrix Of The DerivativeThe case of most interest here is the only one I will discuss. It is the case where X =Rn andY = Rm, the function being defined on an open subset of Rn. Of course this all generalizesto arbitrary vector spaces and one considers the matrix taken with respect to various bases.As above, f will be defined and differentiable on an open set U ⊆ Rn.