MATH 321:201: Real Variables II (Term 2, 2010)
Home work assignment # 10
NOT to be handed-In
Problem 1: Do [Rudin, Ch 9. Exercise # 16 ].
Problem 2:
Let X, Y be metric spaces with metric dX , dY , respectively. Assume the metric space Y is complete. Let 0 < c < 1. Assume f : X × Y → Y is uniformly continuous, i.e. for each > 0
there exists δ > 0 such that for all (x, y), (x0 , y 0 ) ∈ X × Y with dX (x, x0 ) + dY (y, y 0 ) < δ,
dY (f (x, y), f (x0 , y 0 )) ≤ . Assume that for each x ∈ X, the map fx : Y → Y defined as
fx (y) = f (x, y) is Lipschitz with Lipschitz constant c: i.e. dY (fx (y), fx (y 0 )) ≤ cdY (y, y 0 ).
(a) Prove that there is a continuous map g : X → Y such that f (x, g(x)) = g(x) for every x ∈ X.
(b) Assume that X = Rm and Y = Rn and that f is C 1 . Prove that g is C 1 .
Problem 3:(Primitive mapping) Let F : E → Rn be a C 1 mapping on an open set E ⊂ Rn . Assume F (0) = 0 and the derivative DF (0) is invertible, at the origin 0 ∈ Rn , i.e. 0 = (0, 0, · · · , 0).
(a) Prove that there exist open sets U1 , U2 with 0 ∈ Ui , i = 1, 2 and a C 1 diffeomorphism G1 :
U1 → U2 with G1 ((0) = 0, DG1 (0) is invertible, and an invertible n-by-n matrix A such that for
x = (x1 , · · · , xn ), G is of the form
G1 (x) = (g(x), x2 , x3 , · · · , xn )
for some C 1 function g : U1 → R and for
F (x) = AF1 ◦ G1 (x)
where F1 : U2 → Rn is of the form F1 (x) = (x1 , F12 (x), F13 (x), · · · , F1n (x)) for C 1 functions
F1k : U2 → R, k = 2, · · · , n.
(Hint: The proof is an application of the inverse mapping theorem. For more detail, study [Rudin,
10.7])
(b) Use (a) and induction argument to show that for the mapping F as above, there exist opens
neighborhoods U1 , U2 , · · · Un+1 of 0 and mappings G1 , · · · , Gn and invertible matrices A1 , · · · , An−1
such that
F (x) = A1 A2 · · · An−1 Gn ◦ · · · ◦ G1 (x)
where Gk : Uk → Uk+1 , k = 1, · · · n is of the form that the i-th component Gik (x) = xi for i 6= k,
i.e.
Gk (x) = (x1 , x2 , · · · , xk−1 , Gkk (x), xk+1 , · · · , xn )
and G(0) = 0, DGk (0) is invertible. (This form of maps Gk are called primitive mappings.)
The point of this result is that for any C 1 mapping, we can decompose it into more elementary
mappings on a neighborhood where the derivative is invertible.
Problem 4: (complete metric space of differentiable functions and study of nonlinear differential equations) Let the space E of continuously differentiable functions on [0, 1], i.e. E = {f :
[0, 1] → R | f ∈ C 1 on [0, 1]}, be equipped with the norm
kf k1 = sup |f (x)| + sup |f 0 (x)|.
x∈[0,1]
x∈[0,1]
(a) Show that d(f, g) = kf − gk1 gives a metric on E.
(b) Show that (E, d) is a complete metric space. (Hint: Use fundamental theorem of calculus.)
(c) Show that there exists > 0 such that for any continuous function g : [0, 1] → R with
supx∈[0,1] |g(x)| ≤ , there exists a C 1 function f : [0, 1] → R, which solves the following
nonlinear differential equation
(
d
f + f 3 (x) = g(x) ,
dx
(∗) · · ·
f (0) = 0
.
d
f0 = f03 + g, therefore
[Hint: Notice that if f0 is a solution to (∗), then dx
Z x
f0 =
f03 (t) + g(t)dt.
0
i.e f0 satisfies Φg (f0 ) = f0 for the map Φg : E → E defined as
Z x
Φg (f ) =
f 3 (t) + g(t)dt.
0
Note that we cannot directly apply the contraction mapping principle to Φg since Φg itself is not
a contraction! Moreover, for the solution we also require f (0) = 0. So, first we restrict Φg to
E0 = {f ∈ E | f (0) = 0}. Still Φg is not a contraction on E0 . Find a suitable subspace E1 ⊂ E0
such that (1) E1 is a complete metric space; and with a suitable choice of g (i.e. ), (2) Φg : E1 → E1 ;
(3) Φg is a contraction on E1 . ]
The following are suggested exercises. Please DO NOT hand-in, but, it is important for you to
do these suggested exercises!
Problem: Do Rudin, Ch. 9, Exercise # 26
Problem:(Clairaut’s theorem) Study [Rudin, 9.39–41] and do [Rudin, Ch. 9, Exercises # 29].
Problem:(Partition of Unity) Do [Rudin, Ch. 10, Exercises # 5, # 6]
2