Abstarct: The theory of the best approximation is used simultaneously in various branches of mathematics including optimization, numerical analysis, economics and so on. A simple example of this is to find points from a set to a point in space with the least distance. In this ‎thesis‎ we introduce the problem of best approximation, the best simultaneously approximation, the best f-approximation and the best f- simultaneously approximation in different spaces and we are looking for conditions under which the conditions of a proximinal set are simultaneously ‎‎‎‏‎proximinal, f-proximinal, simultaneously f-‎‎‎‏‎proximinal ‎ First introduce best simultaneous approximation in quotient spaces. We give a characterization of best simultaneously approximation and best simultaneously chebyshev in quotient Spaces. Then we introduce the sum of Best simultaneously proximinal ‎subspaces. We introduce the concept of simultaneously ‎‎‎‏‎proximinal in Banach spaces. In the following, we introduce the sum of simultaneously f-proximinal subspaces and Quotient Spaces. We introduce the concept of simultaneously f-proximinal in Banach spaces and the concept of f-simultaneously approximation and we prove some results concerning simultaneously f-approximation of the sum of two subspace in Banach spaces. Furthermore, we introduce the Best Simultaneous Approximation of finite set in Inner product spaces. We introduce the concept on f-Best Approximation in Quotient‎‎ ‎‎‎‎Topological Vector Spaces.