On the Uniform Convexity Of

The standard proof of the uniform convexity of L using Clarkson’s [1] or Hanner’s [2] inequalities (see also [4]) is rarely taught in functional analysis classes, in part (the author imagines) because the proofs of those inequalities are quite non-intuitive and unwieldy. We present here a direct proof, cheerfully sacrificing the optimal bounds – for which, see [2, 4]. It fits quite nicely in with the standard proof of Hölder’s inequality using Young’s inequality Rexy ≤ |x|/p + |y|/q. Recall that equality holds precisely when xy ≥ 0 and |x| = |y|; and that the condition for equality in Hölder’s inequality follows from this. The idea of the present proof is to exploit a lower bound for the difference in Young’s inequality.