Continuous Linear Maps

The continuous linear maps, or operators, are those functions that preserve the structure of normed spaces. They are generalizations of matrices. Various examples on infinite dimensional spaces are given, including the shift operators on sequence spaces and integral operators on the function spaces. Sets of operators itself form normed spaces, a special case being the dual space of a normed space. Examples of calculating the norm of operators are given. The special types of operators called isomorphisms and projections are studied in more detail, and quotient normed spaces are defined and Riesz’s lemma proved. Finally, it is shown that finite dimensional spaces are complete and are the only spaces in which balls are totally bounded.