In this paper we formulate a new structure of spaces which we call it transversal (upper or lower) normed spaces. Combination of algebraic and transversal structures opens up the possibility of studying linear transformation of one transversal normed space into another. This concept of spaces is a natural extension of Banach spaces. Most of our work in this paper centers around forms of three fundamental theorems relating to bounded linear transformations: form of the Hahn-Banach theorem, form of the open mapping theorem and form of the Banach-Steinhaus theorem.