Rotundity and Smoothness

Abstract

When thinking of the closed unit ball of a normed space, it is tempting to visualize some round, smooth object like the closed unit ball of real Euclidean 2- or 3-space. However, closed unit balls are sometimes not so nicely shaped. Consider, for example, the closed unit balls of real ℓ ²₁ and ℓ ²_∞ . Neither is round by any of the usual meanings of that word, since their boundaries, which is to say the unit spheres of the spaces, are each composed of four straight line segments. Also, neither is smooth along its entire boundary, since each has four corners. These features of the closed unit balls have a number of interesting consequences that cause the norms of these two spaces to behave a bit unlike that of real Euclidean 2-space. For example, if z₁ and z₂ are different points on any one of the four sides of one of these balls, then $1\left\| {z_1 } \right\| = \left\| {z_2 } \right\| = \frac{1}{2}\left\| {z_1 } \right\| + \frac{1}{2}\left\| {z_2 } \right\| = \left\| {\frac{1}{2}z_1 + \frac{1}{2}z_2 } \right\|$ so equality is attained in the inequality $\left\| {z_{1 + } z_2 } \right\| \leqslant \left\| {z_1 } \right\| + \left\| {z_2 } \right\|$ despite the fact that neither z₁ nor z₂ is a nonnegative real multiple of the other. Furthermore, the presence of the corners leads to the existence of multiple norming functionals for some points z of the unit sphere of each of these spaces, that is, norm-one members z* of the dual space such that $z^* z = \left\| z \right\|$. To see why this would be, let Z be either of these two spaces and let z₀ be one of the four corners of B _z . Then there are infinitely many different straight lines that pass through z₀ without intersecting the interior of the closed unit ball; let l₁ and l₂ be two of them. By Mazur’s separation theorem, there are members z ^* ₁ and z ^* ₂ of Z*, necessarily different, such that if j ∈ {1, 2}, then z ^* _j z = 1 when z ∈ l _j and z ^* _j z ≤ 1 when z ∈ B _z . It follows readily that z ^* ₁ and z ^* ₁ are both norming functionals for z₀. As will be seen, it is precisely the presence of the corners or sharp bends in the unit sphere that caused this multiplicity of norming functionals for elements at the locations of the bends.