722 CHAPTER 22. THE DERIVATIVE

instead ofDkf(x)(v, · · · ,v) .

Note that for every k, Dkf maps U to a normed vector space. As mentioned above,Df(x) has values in L (X ,Y ) ,D2f(x) has values in L (X ,L (X ,Y )) , etc. Thus it makessense to consider whether Dkf is continuous. This is described in the following definition.

Definition 22.7.2 Let U be an open subset of X , a normed vector space, and let f : U →Y.Then f is Ck (U) if f and its first k derivatives are all continuous. Also, Dkf(x) when it existscan be considered a Y valued multi-linear function. Sometimes these are called tensors incase f has scalar values.

22.8 The Derivative And The Cartesian ProductThere are theorems which can be used to get differentiability of a function based on exis-tence and continuity of the partial derivatives. A generalization of this was given above.Here a function defined on a product space is considered. It is very much like what waspresented above and could be obtained as a special case but to reinforce the ideas, I will doit from scratch because certain aspects of it are important in the statement of the implicitfunction theorem.

The following is an important abstract generalization of the concept of partial derivativepresented above. Insead of taking the derivative with respect to one variable, it is taken withrespect to several but not with respect to others. This vague notion is made precise in thefollowing definition. First here is a lemma.

Lemma 22.8.1 Suppose U is an open set in X×Y. Then the set, Uy defined by

Uy ≡ {x ∈ X : (x,y) ∈U}

is an open set in X. Here X ×Y is a finite dimensional vector space in which the vectorspace operations are defined componentwise. Thus for a,b ∈ F,

a(x1,y1)+b(x2,y2) = (ax1 +bx2,ay1 +by2)

and the norm can be taken to be

||(x,y)|| ≡max(||x|| , ||y||)

Proof: In finite dimensions it doesn’t matter how this norm is defined because all areequivalent. It obviously satisfies most axioms of a norm. The only one which is not obviousis the triangle inequality. I will show this now.

||(x,y)+(x1,y1)|| ≡ ||(x+x1,y+y1)|| ≡max(||x+x1|| , ||y+y1||)≤ max(||x||+ ||x1|| , ||y||+ ||y1||)≤ max(∥x∥ ,∥y∥)+max(∥x1∥ ,∥y1∥)≡ ||(x,y)||+ ||(x1,y1)||