Normed Linear Spaces

In Section 1 we introduce normed linear spaces and bounded linear mappings. Section 2 is concerned with finite-dimensional spaces and with the problem of best approximation by elements of a finite-dimensional subspace. In Section 3 we discuss L_p spaces; we prove the completeness of L_p, and determine the form of the normable linear functionals on L_p in case p> 1. (In contrast to the classical theory, a bounded linear functional need not have a norm.) We then apply these results to the proof of the Radon-Nikodym theorem.