- Suppose L ∈ℒwhere X and Y are two finite dimensional normed vector spaces and suppose L is one to one. Show there exists r > 0 such that for all x ∈ X,
Hint: Show that

≡is a norm. Now suppose L ∈ℒis one to one and onto for X,Y Banach spaces. Explain why the same result holds. Hint: Recall open mapping theorem. - Suppose B is an open ball in X, a Banach space, and f : B → Y is differentiable.
Suppose also there exists L ∈ℒsuch that
for all x ∈ B. Show that if x

_{1},x_{2}∈ B,Hint: Consider Tx = f

− Lx and argue< k. - ↑ Let U be an open subset of X,f : U → Y where X,Y are finite dimensional
normed linear spaces and suppose f ∈ C
^{1}and Dfis one to one. Then show f is one to one near x_{0}. Hint: Show using the assumption that f is C^{1}that there exists δ > 0 such that ifthen

(20.15.42) then use Problem 1. In case X,Y are Banach spaces, assume Df

is one to one and onto. - Suppose U ⊆ X is an open subset of X a Banach space and that f : U → Y is
differentiable at x
_{0}∈ U such that Dfis one to one and onto from X to Y . (Df^{−1}∈ℒ) Then show that f≠ffor all x sufficiently near but not equal to x_{0}. In this case, you only know the derivative exists at x_{0}. - Suppose M ∈ℒwhere X and Y are finite dimensional linear spaces and suppose M is onto. Show there exists L ∈ℒsuch that
where P ∈ℒ

, and P^{2}= P. Also show L is one to one and onto from X_{1}to Y. Hint: Letbe a basis of Y and let Mx_{i}= y_{i}. Then defineShow {x

_{1},,x_{n}} is a linearly independent set and show you can obtain {x_{1},,x_{n},,x_{m}}, a basis for X in which Mx_{j}= 0 for j > n. Then letwhere

- ↑ Let f : U ⊆ X → Y,f is C
^{1}, and Dfis onto for each x ∈ U. Then show f maps open subsets of U onto open sets in Y . Hint: Let P = LDfas in Problem 5. Argue L maps open sets from Y to open sets of X_{1}≡ PX and L^{−1}maps open sets from X_{1}to open sets of Y. Then Lf= Lf+ LDfv + o. Now for z ∈ X_{1}, let h= Lf−Lf. Then h is C^{1}on some small open subset of X_{1}containing 0 and Dh= LDfwhich is seen to be one to one and onto and in ℒ. Therefore, if r is small enough, hequals an open set in X_{1}, V. This is by the inverse function theorem. Hence L= V and so f− f= L^{−1}, an open set in Y. - Suppose U ⊆ ℝ
^{2}is an open set and f : U → ℝ^{3}is C^{1}. Suppose Dfhas rank two andShow that for

near, the points fmay be realized in one of the following forms.or

This shows that parametrically defined surfaces can be obtained locally in a particularly simple form.

- Let f : U → Y , Dfexists for all x ∈ U, B⊆ U, and there exists L ∈ℒ, such that L
^{−1}∈ℒ, and for all x ∈ BShow that there exists ε > 0 and an open subset of B

,V , such that f : V → Bis one to one and onto. Also Df^{−1}exists for each y ∈ Band is given by the formulaHint: Let

for

<, consider {T_{y}^{n}}. This is a version of the inverse function theorem for f only differentiable, not C^{1}. - Denote by Cthe space of functions which are continuous having values in X and define a norm on this linear space as follows.
Show for each λ ∈ ℝ, this is a norm and that C

is a complete normed linear space with this norm. - ↑Let f : × X → X be continuous and suppose f satisfies a Lipschitz condition,
and let x

_{0}∈ X. Show there exists a unique solution to the Cauchy problem,for t ∈

. Hint: Consider the mapdefined by

where the integral is defined componentwise. Show G is a contraction map for

_{λ}given in Problem 9 for a suitable choice of λ and that therefore, it has a unique fixed point in C. Next argue, using the fundamental theorem of calculus, that this fixed point is the unique solution to the Cauchy problem. - ↑Use Theorem 6.7.5 to give another proof of the above theorem. Hint: Use the same mapping and show that a large power is a contraction map.
- Suppose you know that u≤ a + ∫
_{0}^{t}kuds where k≥ 0 and k is in L^{1}. Show that then u≤ aexp. This is a version of Gronwall’s inequality. Hint: Let W= ∫_{0}^{t}kuds. Then explain why W^{′}− kW≤ ak. Now use the usual technique of an integrating factor you saw in beginning differential equations. - ↑Use the above Gronwall’s inequality to establish a result of continuous dependence on the initial condition and f in the ordinary differential equation of Problem 10.
- The existence of partial derivatives does not imply continuity as was shown in an
example. However, much more can be said than this. Consider
Show the directional derivative of f at

exists and equals 0 for every direction. The directional derivative in the directionis defined asNow consider the curve x

^{2}= y^{4}and the curve y = 0 to verify the function fails to be continuous at. - Let
Show that this function is not continuous at

but that it has all directional derivatives atand they all equal 0. - Let X
_{i}be a normed linear space having norm_{i}. Then we can make ∏_{i=1}^{n}X_{i}into a normed linear space by defining a norm on x ∈∏_{i=1}^{n}X_{i}byShow this is a norm on ∏

_{i=1}^{n}X_{i}as claimed. - Suppose f : U ⊆ X × Y → Z and D
_{2}f^{−1}∈ℒexists and f is C^{1}so the conditions of the implicit function theorem are satisfied. Also suppose that all these are complex Banach spaces. Show that then the implicitly defined function y = yis analytic. Thus it has infinitely many derivatives and can be given as a power series as described above.

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