Inequalities

This chapter reviews some real-valued inequalities, and presents the Young's, Holder's, and Minkowski's inequalities, which are elementary. It is used to establish the same inequalities for continuous functions that is used when constructing Lebesgue spaces. The general inequality follows by induction on the dimension d . The chapter establishes a variant of Young's inequality that is related to the weighting operation, which differs from the convolution by a sign. It defines the weighting operation for a continuous function by a real-valued function with compact support. The chapter also presents some properties regarding the integrals of the semi-norms of regularizations for continuous functions, which play a key role when justifying the definition of the semi-norms of Lebesgue spaces.