A new kind of topological vector space: Topological approach vector space

The main goal of the present paper is to introduce the concept of approach topological vector space, which is an extension of topological vector space. In this paper, we explained the relation between metric space and approach space. If we have a contraction function we proved some new properties and present some examples and results in the approach group. We define the approach subgroup and found the relation between topological space and approach space. We found the necessary and sufficient condition to have approach topological space. Also, we define approach vector space, approach subspace, and topological vector space. We introduced a new definition of convergent in approach space and sequentially contraction, we proved contraction and sequentially contraction are equivalent. We proved contraction approach linear map f is sufficient and necessary to have Ker (f) is closed.