A-Best Approximation in Pre-Hilbert C∗-Modules

While there have been many number of studies about best approximation in some spaces, there has been little work on pre-Hilbert C-modules. Here we provide such a study that lead to a number of approximation theorems. In particular, some results about existence and uniqueness of best approximation of submodules on Hilbert C-modules are also presented. This will done by considering the C-algebra valued map x → |x| where |x| = ⟨x; x⟩ 1 2 . Also we show that when K is a convex subset of a pre- Hilbert C-module X; it is a Chebyshev set with respect to C- valued norm which is dened on X. In the end, we study various properties of an A-valued metric projection onto a convex set or a submodule.