This book serves as an introduction to calculus on normed vector spaces at a higher undergraduate or beginning graduate level. The prerequisites include basic calculus and linear algebra, as well as a certain mathematical maturity. All the important topology and functional analysis topics are introduced where necessary.

In its attempt to show how calculus on normed vector spaces extends the basic calculus of functions of several variables, this book is one of the few textbooks to bridge the gap between the available elementary texts and high level texts. The inclusion of many non-trivial applications of the theory and interesting exercises provides motivation for the reader.

Hence the partial derivatives are defined and continuous on Rn nf0g and so the norm k kp is of class C 1 on this set. Case 2. k k1 : all partial derivatives are defined and continuous on the open set S D fx 2 Rn W xi ¤ 0 f or al l i g. We have kxk1 D jx1 j C C jxn j: If xi D 0, then 1 jtj .kx C tei k1 kxk1 / D t t and so @i kxk1 does not exist, which implies that the differential is not defined at a point x with a coordinate whose value is 0. Now suppose that this not the case and that t 2 R is

integral. Let Œa; b be a closed bounded interval of R and E a Banach space. We define a norm k k on B.Œa; b; E/, the vector space of bounded mappings defined on Œa; b with image in E, by setting kf k D sup kf .x/kE : x2Œa;b The normed vector space B.Œa; b; E/ is a Banach space. We say that f W Œa; b ! E is a step mapping if there is a partition P W a D x0 < x1 < < xn D b of Œa; b and elements c1 ; : : : ; cn 2 E such that f .x/ D ci on the interval .xi 1 ; xi /. The set of step mappings,

, we obtain jH 0 .u/j Ä jf 0 .a C h C u/ f 0 .a/ f Œ2 .a/.h C u/j Cj.f 0 .a C u/ f 0 .a/ f Œ2 .a/u/j Cj.f 0 .a C h/ f 0 .a/ f Œ2 .a/h/j: > 0. If h and u are sufficiently small, then we have Let us fix jH 0 .u/j Ä kh C ukE C kukE C khkE Ä 2 .khkE C kukE /: If 2 Œ0; 1, then jH 0 . k/j Ä 2 .khkE C k kkE / Ä 2 .khkE C kkkE / and so sup jH 0 . k/j Ä 2 .khkE C kkkE /: 0Ä Ä1 Therefore k.h; k/ f .2/ .a/.h; k/kF Ä 2 .khkE C kkkE / C khkE kkkE D .3khkE C 2kkkE /kkkE Ä 2 .khkE C kkkE

vector space. A symmetric bilinear form h ; i defined on E is said to be positive definite if for all x 2 E hx; xi 0 and hx; xi D 0 ” x D 0: In this case we say that h ; i is an inner product. The pair .E; h ; i/ is called an inner product space. The dot product on Rn is clearly an inner product and so Rn with the dot product is an inner product space. Exercise 6.1. For M; N 2 Mm;n .R/, let us set hM; N i D tr .N t M /: Show that h ; i defines an inner product on Mm;n .R/. p Proposition 6.1.

interior of A and we write int A for this set. The interior int A is the largest open set lying in A. The intersection of all closed sets containing A is called the closure of A and we write AN for this set. The closure AN is the smallest closed set N The boundary of A, written containing A. Clearly A is closed if and only if A D A. N @A, is the intersection of the sets A and cA. Exercise 1.10. Show that • A is closed if and only if @A A; • AN is the union of A and @A; • a 2 @A if and only if