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 leads 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 be done by considering the c∗-algebra valued map x → |x| where |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 defined on x. in the end, we study various properties of an a-valued metric projection onto a convex set and a submodule.