The main objective to the study of theory of Partial Differential Equations (PDEs) is to insure or find out properties of solutions of PDE that are not directly attainable by direct analytical means. Certain function spaces have certain known properties for which solutions of PDEs can be classified. As a result, this work critically looked into some function spaces and their properties. We consider extensively, Lp-spaces, distribution theory and sobolev spaces. The emphasis is made on sobolev spaces, which permit a modern approach to the study of differential equations. Looking at the linear elliptic partial differential equations considered in this work, we see that the key is Lax-Milgram theorem and the full understanding of Sobolev spaces and its properties. We are able to remove the rigor associated with second order partial differential equations and present it in the form that we can easily handle through the function spaces discussed. The book is based on variational formulation of some Boundary Value Problems (PDEs) using some known theorem (Lax-Milgram Theorem) to ascertain the existence and uniqueness of weak solution to such linear Elliptic PDEs