ON THE SMOOTHNESS OF THE COMPOSITION MAP
ON THE SMOOTHNESS OF THE COMPOSITION MAP
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DOI：
10.1093/qmath/23.2.113
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发表时间：
1972-06
影响因子：
0.7
通讯作者：
M. Irwin
中科院分区：
文献类型：
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作者：
M. Irwin
如果f：E-*· F和y：F->-G是Banach空间的C* 映射，则复合gf：E->-G是C”映射.本文讨论了与复合gf中的（l，g）对相联系的复合映射的光滑性，其中l，g和gf被认为是某些适当的Banach空间中的元素.例如comp：C*-（E，F）x VC+<（F，G）-> &（E，G）是C8，其中UGr+ e（F，G）是从F到G的一致C+* 映射的空间。这种类型的结果是众所周知的，经常使用，但似乎没有令人满意的参考。通常引用的是（1），但情况并不像它看起来的那样微不足道（事实上，（1）的相关定理3.7是错误的，正如下面紧接着推论7的反例所示）。我们在§ 2中讨论了这个理论，并包括一些关于光滑性一致性的讨论，这在应用中经常是有用的。我们给出了三个这样的应用，每个应用都使用了压缩映射定理（在§ 3中通过关于不动点对参数的依赖性的一些结果进行了扩展）。这些是常微分方程的存在定理，哈特曼线性化定理和稳定流形定理。第一个，在§ 4中，基本上是Robbin在（7）中的证明，但是关于初始位置的解的光滑性的证明变得比那里更简单。关于位置和时间的光滑性的证明仍然需要归纳，但我们给出了一个新的直接版本的归纳步骤。我们还讨论了依赖于一个变化的向量场的解决方案。在§ 5中，我们证明了在（6）中证明Hartman线性化定理的Pugh-Moser方法在映射光滑时可以给出额外的信息。简而言之，所得到的共轭性虽然本身不一定是光滑的，但光滑地依赖于线性化的复同态。最后，在§ 6中，我们改进了（3）中给出的稳定流形定理的证明，并将其推广到可以称为相对稳定流形的情形。不幸的是，该方法似乎只能在
IF/: E-*• F and y: F->-G are C* maps of Banach spaces, then the composite gf: E->-G is a C" map. In this paper we discuss smoothness of the composition map, which associates with the pair (/, g) the composite gf, where/, g, and gf are regarded as elements in some appropriate Banach space. For example comp: C*-(E, F) x VC+<(F, G)-> & {E, G) is C8, where UGr+ e (F, G) is the space of uniformly C+* maps from F to G. Results of this type are well known and often used, but there seems to be no satisfactory reference for them. The one usually quoted is (1), but the situation is not quite as trivial as it there appears (in fact the relevant Theorem 3.7 of (1) is false, as the counter-example immediately following Corollary 7 below shows). We run through the theory in § 2, and include some discussion of uniformity of smoothness, which is often useful in applications. We give three such applications, each using the contracting map theorem (extended in § 3 by some results concerning dependence of the fixed point on a parameter). These are to the existence theorem for ordinary differential equations, Hartman's linearization theorem, and the stable manifold theorem. The first, in § 4, is basically Robbin's proof in (7), but the demonstration of smoothness of solutions with respect to initial position alone becomes even simpler than it is there. The proof of smoothness with respect to position and time together still needs an induction, but we give a new direct version of the inductive step. We also discuss dependence of solutions on a varying vector field. In § 5 we show that the Pugh-Moser method in (6) of proving Hartman's linearization theorem for lipeomorphisms may be made to give extra information when the maps are smooth. In brief, the conjugacy obtained, though not itself necessarily smooth, depends smoothly on the diffeomorphism being linearized. Finally in § 6 we improve the proof of the stable manifold theorem given in (3), and generalize it to deal with what might be termed relatively stable manifolds. Unfortunately the method seems only to give smoothness in the