Uniformly smooth space
In mathematics, a uniformly smooth space is a normed vector spacethen The modulus of smoothness of a normed space X is the function ρX defined for every t > 0 by the formula[1] The triangle inequality yields that ρX(t ) ≤ t. The normed space X is uniformly smooth if and only if ρX(t ) / t tends to 0 as t tends to 0.Enflo proved[6] that the class of Banach spaces that admit an equivalent uniformly convex norm coincides with the class of super-reflexive Banach spaces, introduced by Robert C.[7] As a space is super-reflexive if and only if its dual is super-reflexive, it follows that the class of Banach spaces that admit an equivalent uniformly convex norm coincides with the class of spaces that admit an equivalent uniformly smooth norm.The Pisier renorming theorem[8] states that a super-reflexive space X admits an equivalent uniformly smooth norm for which the modulus of smoothness ρX satisfies, for some constant C and some p > 1 It follows that every super-reflexive space Y admits an equivalent uniformly convex norm for which the modulus of convexity satisfies, for some constant c > 0 and some positive real q If a normed space admits two equivalent norms, one uniformly convex and one uniformly smooth, the Asplund averaging technique[9] produces another equivalent norm that is both uniformly convex and uniformly smooth.