208 CHAPTER 8. IMPLICIT FUNCTION THEOREM

Theorem 8.1.6 Suppose U is an open subset of X and f : U → Y is differentiableon U and x+ t (y−x) ∈U for all t ∈ [0,1]. (The line segment joining the two points liesin U.) Suppose also that for all points on this line segment,

∥Df (x+t (y−x))∥ ≤M.

Then∥f (y)−f (x)∥ ≤M |y−x| .

Next recall the following theorem about fixed points of a contraction map. It wasCorollary 3.8.3.

Corollary 8.1.7 Let B be a closed subset of the complete metric space (X ,d) and letf : B→ X be a contraction map

d ( f (x) , f (x̂))≤ rd (x, x̂) , r < 1.

Also suppose there exists x0 ∈ B such that the sequence of iterates { f n (x0)}∞

n=1 remains inB. Then f has a unique fixed point in B which is the limit of the sequence of iterates. Thisis a point x ∈ B such that f (x) = x. In the case that B = B(x0,δ ), the sequence of iteratessatisfies the inequality

d ( f n (x0) ,x0)≤d (x0, f (x0))

1− r

and so it will remain in B ifd (x0, f (x0))

1− r< δ .

The implicit function theorem deals with the question of solving, f (x,y) = 0 for xin terms of y and how smooth the solution is. It is one of the most important theoremsin mathematics. The proof I will give holds with no change in the context of infinite di-mensional complete normed vector spaces when suitable modifications are made on whatis meant by L (X ,Y ) . There are also even more general versions of this theorem than tonormed vector spaces.

Recall that for X ,Y normed vector spaces, the norm on X×Y is of the form

∥(x,y)∥= max(∥x∥ ,∥y∥) .

Theorem 8.1.8 (implicit function theorem) Let X ,Y,Z be finite dimensional normedvector spaces and suppose U is an open set in X ×Y . Let f : U → Z be in C1 (U) andsuppose

f (x0,y0) = 0, D1f (x0,y0)−1 ∈L (Z,X) . (8.2)

Then there exist positive constants, δ ,η , such that for every y ∈ B(y0,η) there exists aunique x(y) ∈ B(x0,δ ) such that

f (x(y) ,y) = 0. (8.3)

Furthermore, the mapping, y→ x(y) is in C1 (B(y0,η)).