Exercise 31, § 1

A normed space E (real or complex) is said to be uniformly convex if, for every $\varepsilon$ such that $0 < \varepsilon < 2$, there exists $\delta > 0$ such that the relations $\|x\| \leq 1, \|y\| \leq 1, \|x - y\| \geq \varepsilon$ in E imply $\|\frac{1}{2}(x + y)\| \leq 1 - \delta$. A uniformly convex space is strictly convex (V, p. 67, exerc. 27). We say that E is uniformly smooth if, for every $\varepsilon > 0$, there exists $\eta > 0$ such that the relations $\|x\| \geq 1, \|y\| \geq 1, \|x - y\| \leq \eta$ imply the inequality $\|x + y\| \geq \|x\| + \|y\| - \varepsilon \|x - y\|$. This is equivalent to : for every $\varepsilon > 0$, there exists $\rho > 0$ such that the relations $\|x\| = 1, \|y\| \leq \rho$ imply the inequality

A uniformly smooth space is smooth (V, p. 66, exerc. 26).

a) Show that if E is uniformly convex, its strong dual E' is uniformly smooth, and that if E is uniformly smooth, then E' is uniformly convex; the restriction of the unique support mapping (V, p. 67, exerc. 29) to the unit sphere S of E is a mapping from S into the unit sphere S' of E', which is continuous for the norm topologies of E and E'. b) Show that if E is uniformly convex, and, if a filter $\mathfrak{F}$ on E converges to $x_0$ for the topology $\sigma(E, E')$ and is such that $\lim_{\mathfrak{F}} \|x\| = \|x_0\|$, then $\mathfrak{F}$ converges to $x_0$ for the initial topology of E. c) Show that a Banach space which is uniformly convex or uniformly smooth is reflexive (use b) and c) and also IV, p. 60, exerc. 12). (cf. V, p. 72, exerc. 14.) d) Generalize the first part of th. 1 of V, p. 10, and also cor. 1 and 2 of V, p. 11 to uniformly convex Banach spaces.