New Generalization of -Best Simultaneous Approximation in Topological Vector Spaces

Let K be a nonempty subset of a Hausdorff topological vector space X, and let f be a real-valued continuous function on X. If for each x = (x(1), x(2), ..., x(n)) is an element of X-n, there exists k(0) is an element of K such that F-K(x) = Sigma(n)(i=1) f(x(i) - k(0)) = inf{Sigma(n)(i=1) f(x(i) - k) : k is an element of K}, then K is called f-simultaneously proximal and k(0) is called f-best simultaneous approximation for x in K. In this paper, we study the problem of f-simultaneous approximation for a vector subspace K in X. Some other results regarding f-simultaneous approximation in quotient space are presented.