This paper represents the theory leading up to the Representation theorem of a linear functional, which we will use later to introduce the Hermitian adjoint operators. The paper first introduces unitary and normed vector spaces and mentions some terms accompanying them such as orthogonal vectors, orthogonal complements and the norm of a vector. Afterwards we will define linear and antilinear operators, which are closely related, and analyze the differences between them. Furthermore, we will explain what monomorphisms, epimorphisms and isomorphisms are, and mention different criteria and additional conditions for identifying them or even make them indistinguishable. We will also consider conjugate spaces of a vector space and, through examples, touch upon the subject of linear functionals and state the Riesz representation theorem which enables us to have a unique depiction of a functional. The end of this paper will be reserved for the explanation of the Hermitian adjoint and its properties.