Chapter 9 - The Riesz Theorem

Abstract

This chapter discusses about the Riesz theorem. The use of the Riesz theorem that is used in tandem with the Hahn–Banach theorem is examined in the chapter. One of the first applications of the Riesz theorem is in characterizing weakly convergent sequences in metric spaces. The partnership of the Riesz Theorem with various forms of the Hahn–Banach theorem is the most powerful in all abstract analysis. The most basic form of the Hahn–Banach theorem extending bounded linear functional from a subspace of a Banach space to the whole space without changing the norm is analyzed in the chapter. The Banach's version of the extension theorem is applied wherein a linear functional that is dominated by a positively homogeneous, subadditive functional is extended to the whole space with domination remaining. One remarkable consequence of Choquet's theorem is the characterization of weakly null sequences in Banach spaces. It is shown that between Hilbert spaces, the 2-summing operators and Hilbert–Schmidt operators are precisely the same, while the trace-class coincides with the nuclear operators.