By J. T. Schwartz

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If it is, then the differential of fi = df will have its values in the space B2 (X, Z) of bounded bilinear functions of X to Z, where, by the lemma above, we have identified B2 (X, Z) and B (X, B (X, Z)). We define the differential of fi = df at a point c to be the second differential off at c and we denote this second differential by PA x). Hence d2f (c) is a bounded bilinear function on X to Z. Higher order differentials are defined by induction. 42. Defuidon: A function f on D c X to Z is said to be in class C" on D, written f e C", iff the n-th differential d"f exists at every point of D and the mapping x - d"lx) of D into B" (X; Z) is continuous.

Theorem: Suppose that f e Cl on an open set D which contains the line segment joining c to c + x. Then 29 BASIC CALCULUS f(c + x) = f(c) + 1! df(c; x) + + (n-I)! + 1 1 d2f(c; x(2)) + ... 21 d"-lf(c; x("-") - t)"-1 d"f(c + tx; dt. Proof: fo Since the map t -+ d*Ac + tx; x(")) is continuous on [0, 1) to Z, it is clear that both sides of the equation have a meaning. To establish the equality let Z* be a continuous linear functional on Z and let F be de- fined on 10, 11 to the scalar field by F(t) = Z*f (c + tx).

J. Nash, "The imbedding problem for Riemannian manifolds", Ann. of Math. vol. 63, pp. 20-63 (1956). Now we shall consider the problem of isometric embeddings of Riemannian manifolds in euclidean spaces. 2 above. The problem can be stated as follows. Is every Riemannian manifold (say of class Ck) isometrically embeddable in RI? ) Nash's answer is in the affirmative (technically, when k >- 3), and he also asserts that m may be chosen less than or equal to an explicit function of the dimension n of the manifold (namely m 5 1 (3n3 + 14n2 + 11 n) for the general case and m 5 1 n(3n + 11) if M is compact).

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