Chapter 4

The DerivativeThe derivative is a linear transformation. In this chapter are the principal results about thederivative.

4.1 Basic DefinitionsThe derivative is a linear transformation. This may not be entirely clear from a beginningcalculus course because they like to say it is a slope which is a number. As observed byDeudonne,

“...In the classical teaching of Calculus, this idea (that the derivative is a lineartransformation) is immediately obscured by the accidental fact that, on a one-dimensional vector space, there is a one-to-one correspondence between linearforms and numbers, and therefore the derivative at a point is defined as a num-ber instead of a linear form. This slavish subservience to the shibboleth1 ofnumerical interpretation at any cost becomes much worse when dealing withfunctions of several variables...”

The concept of derivative generalizes right away to functions of many variables but onlyif you regard a number which is identified as the derivative in single variable calculus as alinear transformation on R. However for functions of many variables, no attempt will bemade to consider derivatives from one side or another. This is because when you considerfunctions of many variables, there isn’t a well defined side. However, it is certainly the casethat there are more general notions which include such things. I will present a fairly generalnotion of the derivative of a function which is defined on an open subset of a normed vectorspace which has values in a normed vector space. The case of most interest is that of afunction which maps an open set in Fn to Fm but it is no more trouble to consider the extragenerality and it is sometimes useful to have this extra generality because sometimes youwant to consider functions defined, for example on subspaces of Fnand it is nice to nothave to trouble with ad hoc considerations. Also, you might want to consider Fn with somenorm other than the usual one.

For most of what follows, it is not important for the vector spaces to be finite dimen-sional provided you make the following definition of what is meant by L (X ,Y ) which isautomatic if X is finite dimensional. See Proposition 2.8.8.

Definition 4.1.1 Let (X ,∥·∥X ) and (Y,∥·∥Y ) be two normed linear spaces. ThenL (X ,Y ) denotes the set of linear maps from X to Y which also satisfy the following con-dition. For L ∈L (X ,Y ) ,

lim∥x∥X≤1

∥Lx∥Y ≡ ∥L∥< ∞

To save notation, I will use ∥·∥ as a norm on either X , Y or L (X ,Y ) and allow thecontext to determine which it is.

Let U be an open set in X , and let f : U → Y be a function.1In the Bible, there was a battle between Ephraimites and Gilleadites during the time of Jepthah, the judge

who sacrificed his daughter to Jehovah, one of several instances of human sacrifice in the Bible. The cause ofthis battle was very strange. However, the Ephramites lost and when they tried to cross a river to get back home,they had to say shibboleth. If they said “sibboleth” they were killed because their inability to pronounce the “sh”sound identified them as Ephramites. They usually don’t tell this story in Sunday school. The word has come todenote something which is arbitrary and no longer important.

91