New Generalization of f -Best Simultaneous Approximation in Topological Vector Spaces

Abstract

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 $x=(x_1,x_2,\dots ,x_n)\in X^n$, there exists $k_0\in K$ such that $F_K(x)={\sum }_{\mathrm{i}=1}^{\mathrm{n}}f\left(x_i-k_0\right)=\inf \left\{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}f(x_i-k):k\in K\right\}$, 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.