Basic Inequalities and Lebesgue Spaces

Abstract

I have already introduced (cf. §§ 3.1.2 and 3.2.3) the vector space $L^1(\mu ;{\mathbb R})$ with the norm $\Vert {\,\cdot \,}\Vert _{L^1(\mu ;{\mathbb R})}$ and shown it to be a Banach space: that is, a normed vector space that is complete with respect to the metric determined by its norm. Although, from the measure-theoretic point of view, $L^1(\mu ;{\mathbb R})$ is an natural space with which to deal, from a geometric standpoint, it is flawed. To understand its flaw, consider the two point space $E=\{1,2\}$ and the measure $\mu$ that assigns measure 1 to both points. Then $L^1(\mu ;{\mathbb R})$ is easily identified with ${\mathbb R}^2$, and the length that $\Vert {\,\cdot \,}\Vert _{L^1(\mu ;{\mathbb R})}$ assigns $x=(x_1,x_2)\in {\mathbb R}^2$ is $|x_1|+|x_2|$. Hence, the unit ball in this space is the equilateral diamond whose center is the origin and whose vertices lie on the coordinate axes, and, as such, its boundary has nasty corners. For this reason, it is reasonable to ask whether there are measure-theoretically natural Banach spaces that have better geometric properties.