Unique best nonlinear approximation in Hilbert spaces

Source 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 H be a Hilbert space and F F map R n {R^n} into H H such that F F is a homeomorphism onto F = F ( R n ) \mathfrak {F} = F({R^n}) and is twice continuously Fréchet differentiable. Then if F ′ ( α ) ⋅ R n F’(\alpha ) \cdot {R^n} is of dimension n n for all α ∈ R n \alpha \in {R^n} , the manifold F \mathfrak {F} has finite curvature everywhere. It follows that there is a neighborhood U \mathfrak {U} of F \mathfrak {F} such that each u ∈ U u \in \mathfrak {U} has a unique best approximation from F \mathfrak {F} . However, these results do not hold in general for uniformly smooth Banach spaces.