By Cornelius Lanczos
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Sample text
Suppose fECI (M, ~ 1 ). Let N' c N be two closed neighborhoods satisfying dist(N',8N) 2: ~8, 8> o. Suppose that there are constants band f: positive, such that IWex)1I 2: b 'v' x E fc+< \(fc-< UN'), 0< f < Min {~8b2, ~8b}. 3. 3. According to (PS)c, Kc is compact. Hence, for 8> 0 sufficiently small, N(8) = {x E Mldist(x,Kc) < 8} eN. 4, if we take N' = N(~). 4. Define a smooth function: pes) = { ~ for s rt. [e - f:, c + f:J, for s E [e - f, C + f], with 0::; pes) ::; 1. Let A = M\(N'h, where (N')o = {x E Mldist(x, N') ::; 8 8}, and B = N' be two closed subsets.
E. Combining T/ with ~, we obtain the deformation retract. 3) is easily verified. This completes the proof. 4 are Milnor [Mill], Schwartz [ScJl], Rothe [Rotl], Palais [Pall], Pitcher [Pitl] and Marino Prodi [MaPl]. 4, the handle body theorem is established on Hilbert Riemannian manifolds, where the Morse Lemma holds, and the local behavior of a nondegenerate critical point is quite clear. In order to extend this theorem to Finsler manifolds, or to Banach spaces, new difficulties arise in two ways.
In fact (1), (3), and (4) follow directly from the construction. (2) holds if IIbll is small. Furthermore, 9 differs from f only in the neighborhood cp-l B(9, 6). Since Ho n B(9,6) is compact (dimHo = dimker~ f(p) < +00) and d 2 g 0 cp-l I(H+ffiH_)nB(8,6) is invertible, we conclude that 9 satisfies the (PS) condition if f does. 6. 1, in which we assumed the negative gradient points inward at the boundary. The purpose of this section is to extend the previous study in two aspects: (1) under more general boundary conditions, and (2) when the underlying space is not a manifold but a locally convex closed set.