196 CHAPTER 7. THE DERIVATIVE

then (u,y) ∈U. This has just said that B(x,r)X , the ball taken in X is contained in Uy .This proves the lemma. ■

Or course one could also consider Ux≡{y : (x,y) ∈U} in the same way and concludethis set is open in Y . Also, the generalization to many factors yields the same conclusion.In this case, for x ∈∏

ni=1 Xi, let

∥x∥ ≡max(∥xi∥Xi

: x= (x1, · · · ,xn))

Then a similar argument to the above shows this is a norm on ∏ni=1 Xi. Consider the triangle

inequality.

∥(x1, · · · ,xn)+(y1, · · · ,yn)∥= maxi

(∥xi +yi∥Xi

)≤max

i

(∥xi∥Xi

+∥yi∥Xi

)≤max

i

(∥xi∥Xi

)+max

i

(∥yi∥Xi

)= ∥x∥+∥y∥

Corollary 7.9.2 Let U ⊆∏ni=1 Xi be an open set and let

U(x1,··· ,xi−1,xi+1,··· ,xn) ≡ {x ∈ Fri : (x1, · · · ,xi−1,x,xi+1, · · · ,xn) ∈U} .

Then U(x1,··· ,xi−1,xi+1,··· ,xn) is an open set in Fri .

Proof: Let z ∈U(x1,··· ,xi−1,xi+1,··· ,xn). Then (x1, · · · ,xi−1,z,xi+1, · · · ,xn)≡ x ∈U bydefinition. Therefore, since U is open, there exists r > 0 such that B(x,r)⊆U. It followsthat for B(z,r)Xi

denoting the ball in Xi, it follows that B(z,r)Xi⊆U(x1,··· ,xi−1,xi+1,··· ,xn)

because to say that ∥z−w∥Xi< r is to say that

∥(x1, · · · ,xi−1,z,xi+1, · · · ,xn)− (x1, · · · ,xi−1,w,xi+1, · · · ,xn)∥< r

and so w ∈U(x1,··· ,xi−1,xi+1,··· ,xn). ■Next is a generalization of the partial derivative.

Definition 7.9.3 Let g : U ⊆∏ni=1 Xi→ Y , where U is an open set. Then the map

z→ g (x1, · · · ,xi−1,z,xi+1, · · · ,xn)

is a function from the open set in Xi,

{z : x= (x1, · · · ,xi−1,z,xi+1, · · · ,xn) ∈U}

to Y . When this map is differentiable, its derivative is denoted by Dig (x). To aid in thenotation, for v ∈ Xi, let θ iv ∈∏

ni=1 Xi be the vector (0, · · · ,v, · · · ,0) where the v is in the

ith slot and for v ∈∏ni=1 Xi, let vi denote the entry in the ith slot of v. Thus, by saying

z→ g (x1, · · · ,xi−1,z,xi+1, · · · ,xn)

is differentiable is meant that for v ∈ Xi sufficiently small,

g (x+θ iv)−g (x) = Dig (x)v+o(v) .

Note Dig (x) ∈L (Xi,Y ) .