We prove that every continuous mapping from a separable infinite-dimensionalHilbert space $X$ into $\mathbb{R}^{m}$ can be uniformly approximated by$C^\infty$ smooth mappings {\em with no critical points}. This kind of resultcan be regarded as a sort of very strong approximate version of the Morse-Sardtheorem. Some consequences of the main theorem are as follows. Every twodisjoint closed subsets of $X$ can be separated by a one-codimensional smoothmanifold which is a level set of a smooth function with no critical points;this fact may be viewed as a nonlinear analogue of the geometrical version ofthe Hahn-Banach theorem. In particular, every closed set in $X$ can beuniformly approximated by open sets whose boundaries are $C^\infty$ smoothone-codimensional submanifolds of $X$. Finally, since every Hilbert manifold isdiffeomorphic to an open subset of the Hilbert space, all of these resultsstill hold if one replaces the Hilbert space $X$ with any smooth manifold $M$modelled on $X$.
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机译：我们证明，从可分离的无限维希尔伯特空间$ X $到$ \ mathbb {R} ^ {m} $的每个连续映射都可以由$ C ^ \ infty $平滑映射{\ em没有临界点}统一近似。这种结果可以看作是莫尔斯·萨德定理的一种非常强的近似形式。主定理的一些结果如下。 $ X $的每两个不相交的闭合子集可以由一个一维平滑流形分隔，该平滑流形是没有临界点的平滑函数的水平集；这一事实可以看作是Hahn-Banach定理的几何形式的非线性类似物。特别地，$ X $中的每个闭合集都可以用边界为$ X $的$ C ^ \ infty $光滑子维子流形的开放集统一逼近。最后，由于每个希尔伯特流形与希尔伯特空间的一个开放子集是异形的，因此，如果用在$ X $上建模的任何光滑歧管$ M $替换希尔伯特空间$ X $，所有这些结果仍然成立。
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