Number Theory: An Introduction to Mathematics

34 I The Expanding Universe of Numbers

We will also use the notion of norm of a linear map. IfA:Rn→Rmis a linear

map, itsnorm|A|is defined by

|A|=sup

|x|≤ 1

|Ax|.

Evidently

|A 1 +A 2 |≤|A 1 |+|A 2 |.

Furthermore, ifB:Rm→Rlis another linear map, then

|BA|≤|B||A|.

Hence, ifm=nand|A|<1, then the linear mapI−Ais invertible, its inverse being

given by the geometric series

(I−A)−^1 =I+A+A^2 +···.

It follows that for any invertible linear mapA:Rn→Rn,ifB:Rn→Rnis a lin-

ear map such that|B−A|<|A−^1 |−^1 ,thenBis also invertible and|B−^1 −A−^1 |→ 0

as|B−A|→0.

Ifφ:U→Rmis differentiable atx 0 ∈Rn, then it is also continuous atx 0 ,since

|φ(x)−φ(x 0 )|≤|φ(x)−φ(x 0 )−φ′(x 0 )(x−x 0 )|+|φ′(x 0 )||x−x 0 |.

We s a y t h a tφiscontinuously differentiableinUif it is differentiable at each point of
Uand if the derivativeφ′(x)is a continuous function ofxinU.Theinverse function
theoremsays:
Proposition 27Let U 0 be a neighbourhood of x 0 ∈Rnand letφ:U 0 →Rnbe a
continuously differentiable map for whichφ′(x 0 )is invertible.
Then, for someδ> 0 , the ball U={x∈Rn:|x−x 0 |<δ}is contained in U 0
and
(i)the restriction ofφto U is injective;
(ii)V:=φ(U)is open, i.e. ifη∈V , then V contains all y∈Rnnearη;
(iii)the inverse mapψ:V→U is also continuously differentiable and, if y=φ(x),
thenψ′(y)is the inverse ofφ′(x).

Proof To simplify notation, assumex 0 =φ(x 0 )=0 and writeA=φ′( 0 ).Forany

y∈Rn, put

fy(x)=x+A−^1 [y−φ(x)].

Evidentlyxis a fixed point offyif and only ifφ(x)=y.Themapfyis also contin-

uously differentiable and

fy′(x)=I−A−^1 φ′(x)=A−^1 [A−φ′(x)].

Sinceφ′(x)is continuous, we can chooseδ>0 so that the ballU={x∈Rn:|x|<δ}

is contained inU 0 and

|fy′(x)|≤ 1 /2forx∈U.