Uniformly smooth space

In mathematics, a uniformly smooth space is a normed vector space X satisfying the property that for every \epsilon>0 there exists \delta>0 such that if x,y\in X with \|x\|=1 and \|y\|\leq\delta then

\|x+y\|+\|x-y\| \le 2 + \epsilon\|y\|.

The modulus of smoothness of a normed space X is the function ρX defined for every t > 0 by the formula[1]

\rho_X(t) = \sup \Bigl\{ \frac{\|x + y \| + \|x - y\|}{2} - 1 \,:\, \|x\| = 1, \; \|y\| = t \Bigr\}.

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.

Properties

\rho_{X^*}(t) = \sup \{ t \varepsilon / 2 - \delta_X(\varepsilon) : \varepsilon \in [0, 2]\}, \quad t \ge 0,
and the maximal convex function majorated by the modulus of convexity δX is given by[4]
\tilde \delta_X(\varepsilon) = \sup \{ \varepsilon t / 2 - \rho_{X^*}(t) : t \ge 0\}.
Furthermore,[5]
\delta_X(\varepsilon / 2) \le \tilde \delta_X(\varepsilon) \le \delta_X(\varepsilon), \quad \varepsilon \in [0, 2].
\lim_{t\to 0}\frac{\|x+ty\|-\|x\|}{t}
exists uniformly for all x, y\in S_X (where S_X denotes the unit sphere of X).

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. James.[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

\rho_X(t) \le C \, t^p, \quad t > 0.

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

\delta_Y(\varepsilon) \ge c \, \varepsilon^q, \quad \varepsilon \in [0, 2].

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.

See also

Notes

  1. ↑ see Definition 1.e.1, p. 59 in Lindenstrauss & Tzafriri (1979).
  2. ↑ Proposition 1.e.3, p. 61 in Lindenstrauss & Tzafriri (1979).
  3. ↑ Proposition 1.e.2, p. 61 in Lindenstrauss & Tzafriri (1979).
  4. ↑ Proposition 1.e.6, p. 65 in Lindenstrauss & Tzafriri (1979).
  5. ↑ Lemma 1.e.7 and 1.e.8, p. 66 in Lindenstrauss & Tzafriri (1979).
  6. ↑ Enflo, Per (1973), "Banach spaces which can be given an equivalent uniformly convex norm", Israel J. Math. 13:281–288.
  7. ↑ James, Robert C. (1972), "Super-reflexive Banach spaces", Canad. J. Math. 24:896–904.
  8. ↑ Pisier, Gilles (1975), "Martingales with values in uniformly convex spaces", Israel J. Math. 20:326–350.
  9. ↑ Asplund, Edgar (1967), "Averaged norms", Israel J. Math. 5:227–233.

References