A Banach space E is defined as cP,q smooth if Cp'q(E,R) contains a nonzero function with bounded support. If f E Cp'q(E,F), if F Cp,q is smooth and if E is non-Cp,q smooth, then the image under f of the boundary of any bounded open subset U of E is dense in the image of U. If E is separable, then E Cp,q is smooth if and only if E Cp,q admits partitions of unity; Claim cP,Psmooth,p<00,if.
Show that if E is separable and Cp,q smooth and if f E Cq(E,F) is C approximate in some neighborhood. We define a space B to be Cp,q smooth if there exists a nonzero function on cP,q(E,R) with bounded support. One property in particular is that a map into Cp,q(E,F) has an "analytic" property if F is cP,q smooth and E is not Cp,q smooth: the values of Fon a bounded open subset U.
Second, if E is only cP smooth and A is a closed subset of E, we provide a sufficient condition for A to ensure that it is the zero set of some cP function. Bonic and Frampton in [l] proved that if E is separable, then E admits cP partitions of unity if and only if E is. If f is a map from a Banach space E to a Banach space F, f is said to have a Gateaux derivative at x in the direction h if GDf(x)[h] = lim.
Proposition l._2 If f:E-F has a Frechet derivative at all points on the segment [x,x+h], then for any wEF*.
CHAPTER II
In the next chapter we will prove that there exists a c 2 smooth B-space that is not c2' 2 smooth. If E Cp,q is smooth, then every B-space equivalent to E and every closed subspace of E is again Cp,q smooth. Theorem 2.2 A Banach space E is Cp,q smooth if and only if the norm topology on E is equivalent to the topology induced on E by the functions cP,q(E,R).
Thus the functions Cp,q in a smooth non-Cp,q space B have a kind of "analytic" property: the values of the function in a bounded open set are uniquely determined by the values on the boundary. If p i is an odd integer, pP and thus any infinite dimensional LP space is not nP smooth. Finally, if p is not an integer, then f P and any infinite dimensional LP space is not C~~EpJ smooth.
E has a partition cP of unity (cpa.} such that the support of every cp a. is contained in some U13. It is not known whether divisibility can be removed by Prop.2.6, in particular, it is not known whether every indivisible space Hilbert accepts c1 partitions of unity.l) Refer to Bonic and Frampton [l].
CHAPTER III
Since a.(X) locally depends on only a limited number of variables, we can apply the finite-dimensional implicit function theorem to conclude that a.(X) E. The following theorem will imply that a collision function C2 '2 could not be found for c. 0 • We prove a slightly stronger result. 2N components of X have absolute value l/N, the remainder. the component of the first 2N components has an absolute value less than or equal to l/N and all other components are zero.
IJ.1 Therefore we can choose inductively; y H. at least 2 components equal to k N. 0 ,R) has a uniformly continuous first derivative.
CHAPTER IV
Then there exists a closed subset of E that is not the null set of any cP function. By Theorems 4.1 and 4.2, a separable Banach space, E, cP,P is smooth if and only if every closed subset of E is the null set of a cP function. If every closed subset of a non-separable Hilbert space H were the null set of a cP.
A whose complement has a cylindrical bou!ldary is the zero set of some cP function. Then the complement of A is the zero set of a. the proof of Theorem 4.1 to show that the derivatives of all the partial sums converge uniformly. Let U be any neighborhood of zero and suppose that U contains the open ball of radius r.
Find n such that l/n < r/2 and let W be a weak neighborhood of e1/n,([ei} is the orthonornal basis). In this chapter we show that a separable cP,q smooth B-space Cp,q allows unit partitions. Then (Ax} includes E and since E is Lindelof, we can extract a countable subset of (Ax}, which too.
To show that (Vi} includes E, suppose that x EE and that i(x) is the first integer for which. Then, by the continuity of cpn(x)', there exists a neighborhood Nx of x and an ax > ~ such that If E Cp,q is smooth and separable, then using the above theorem we can find a Cp,q-partition of unity [ c µ.
In fact, if the supports of the partition functions have uniformly bounded diameters, then by Lebesque's Covering Theorem for any N there are points of E where more than N of the partition functions are non-zero.
CHAPTER VI SMOOTH APPROXIMATION
The theorem below will show that if a Cq function on a separable Cp,q smooth B-space is locally cp,q approximable, then i t is strongly approximable on the entire space. This would be an essential theorem in the construction of C p,q approximations on manifolds modeled on Cp,q smooth Banach spaces. Let f E Cq(E,F) and assume that for each x in E there is a neighborhood Nx of x such that f is C that can be approximated on Nx.
V~} and tv{l cover E, for each x there is a neighborhood U of x and an integer kx such that f. We see that cr(x) is not differentiable anywhere. • To show this, let's take x E p2 and assume that cr is differentiable in x. We ask the question: is there a better c2 l approximation of L(x) on the unit ball than a constant.
CHAPTER VII
In the subsequent constructions, we need two propositions about measures on Banach spaces. In the proof of the next theorem, we will use the measures µA to form cP functions of 1 2 • We remember that. Then S is positive definite self-adjoint Hilbert-Schmidt and if we denote the unit sphere by B, then
Then the conclusion of the theorem would be true if the operator •r were only assumed to be. The proof is similar to the proof of Theorem 6.1 in which Lemma 5.1 played a key role.
