ABSTRACT
This chapter introduces many new ideas and studied. The reader is advised to give special attention to definitions; not only should every definition be memorized, but sufficient time should be spent to ensure that a solid intuitive grasp of each new concept is attained. In a very general way, continuous functions bear the same relationship to metric spaces that linear transformations do to vector spaces. The chapter shows that linear transformations are of special importance because they take the algebraic structure of vector spaces into account. The reader should appreciate the natural parallel between vector spaces and metric spaces: in the case of vector spaces, the algebraic structure is in some sense preserved by linear transformations while in the case of metric spaces the convergence of sequences is maintained.
