Smooth isometric immersions into the infinite-dimensional sphere (Q1974253)

This paper deals with uniformly smooth immersions and embeddings of infinite-dimensional Banach spaces into arbitrary neighborhoods of zero. In the case of a separable Hilbert space \(X\) the authors construct a uniformly smooth isometric embedding of \(X\) into its unit sphere \(S\). They next state two similar results for the case where \(X\) is a separable Banach space and \(S\) is the unit sphere of \(C([0,1])\), respectively of \(L_p([0,1]),~~p\in [1,\infty]\).