1. Homework 1
(1) Let (X, k·kX) and (Y, k·kY) be normed vector spaces over R. A linear map T : X → Y is a bounded linear map if there exists M > 0 such that
kT (x)kY ≤ M kxkX.
Prove that the following three statements are equivalent.
1. T : X → Y is bounded.
2. T : X → Y is continuous.
3. T : X → Y is continuous at 0.
Hint. To prove (3. implies 1.), you assume that T is continuous at 0. Given
= 1, you choose δ > 0 so that kT (x)kY < 1 whenever kxkX < δ. If w ∈ X with w 6= 0, denote x = δw
2kwkX. Then kxkX < δ.
(2) Let f : R2→ R be the function f (x, y) =
xy2
x2+ y4 if (x, y) 6= (0, 0), 0 if (x, y) = (0, 0).
Prove that the directional derivatives of f exist at (0, 0) in all direction u ∈ R2 with kuk = 1, i.e. Duf (0, 0) exist for all u ∈ R2 with kuk = 1 and f is not continuous at (0, 0).
(3) Let f : U ⊂ Rn → Rm be a function defined in an open subset U of Rn containing the origin 0. Suppose that there exists M > 0 and δ > 0 such that B(0, δ) ⊆ U and
kf (x)k ≤ M kxk2 for any x ∈ B(0, δ).
Show that f is differentiable at 0 and find Df (0).
(4) Let f : R → R be the function f (x) = x sin x. Compute Df (x) and df dx. Remark. You will understand the difference between Df (x) and df
dx after doing this exercise.
(5) Let f, g : U ⊂ Rn → Rm be two functions defined on an open subset U of Rn. Assume that f and g are both differentiable at p ∈ U. Prove that
D(f + g)(p) = Df (p) + Dg(p).
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