Unique best nonlinear approximation in Hilbert spaces

Abstract:

Using the notion of curvature of a manifold, developed by J. R. Rice and recently studied by E. R. Rozema and the second named author, the authors prove the following result: Let $H$ be a Hilbert space and $F$ map ${R^n}$ into $H$ such that $F$ is a homeomorphism onto $\mathfrak {F} = F({R^n})$ and is twice continuously Fréchet differentiable. Then if $F’(\alpha ) \cdot {R^n}$ is of dimension $n$ for all $\alpha \in {R^n}$, the manifold $\mathfrak {F}$ has finite curvature everywhere. It follows that there is a neighborhood $\mathfrak {U}$ of $\mathfrak {F}$ such that each $u \in \mathfrak {U}$ has a unique best approximation from $\mathfrak {F}$. However, these results do not hold in general for uniformly smooth Banach spaces.