14.5. EXERCISES 345

12. ↑ Now for p > 1, verify that ∥x+y∥p ≤ ∥x∥p +∥y∥p . Then verify the other axiomsof a norm. This will give an infinite collection of norms for Fn. Hint: You might dothe following.

∥x+y∥pp ≤

n

∑k=1|xk + yk|p−1 (|xk|+ |yk|)

=n

∑k=1|xk + yk|p−1 |xk|+

n

∑k=1|xk + yk|p−1 |yk|

Now explain why p−1 = p/q and use the Holder inequality.

13. This problem will reveal the best kept secret in undergraduate mathematics, the defi-nition of the derivative of a function of n variables. Let ∥·∥ be a norm on Fn and alsodenote by ∥·∥ a norm on Fm. If you like, just use the standard norm on both Fn andFm. It can be shown that this doesn’t matter at all (See Problem 25 on 422.) but toavoid possible confusion, you can be specific about the norm. A set U ⊆ Fn is saidto be open if for every x ∈U, there exists some rx > 0 such that B(x,rx)⊆U where

B(x,r)≡ {y ∈ Fn : ∥y−x∥< r}

This just says that if U contains a point x then it contains all the other points suffi-ciently near to x. Let f : U 7→ Fm be a function defined on U having values in Fm.Then f is differentiable at x ∈U means that there exists an m×n matrix A such thatfor every ε > 0, there exists a δ > 0 such that whenever 0 < ∥v∥< δ , it follows that

∥f(x+v)− f(x)−Av∥∥v∥

< ε

Stated more simply,

lim∥v∥→0

∥f(x+v)− f(x)−Av∥∥v∥

= 0

Show that A is unique and verify that the ith column of A is

∂ f∂xi

(x)

so in particular, all partial derivatives exist. This unique m× n matrix is called thederivative of f. It is written as Df(x) = A.