Inmathematics, auniformly smooth space is anormed vector space${\displaystyle X}$ satisfying the property that for every${\displaystyle ϵ>0}$ there exists${\displaystyle \delta >0}$ such that if${\displaystyle x,y\in X}$ with${\displaystyle \Vert x\Vert =1}$ and${\displaystyle \Vert y\Vert \le \delta }$ then

${\displaystyle \Vert x+y\Vert +\Vert x-y\Vert \le 2+ϵ\Vert y\Vert .}$

Themodulus of smoothness of a normed spaceX is the function ρ_X defined for everyt > 0 by the formula^[1]

${\displaystyle {\rho }_X(t)=sup\{\frac{\Vert x+y\Vert +\Vert x-y\Vert }{2}-1:\Vert x\Vert =1,\Vert y\Vert =t\}.}$

The triangle inequality yields thatρ_X(t ) ≤t. The normed spaceX is uniformly smooth if and only ifρ_X(t ) /t tends to 0 ast tends to 0.

• Every uniformly smoothBanach space isreflexive.^[2]
• A Banach space${\displaystyle X}$ is uniformly smooth if and only if itscontinuous dual${\displaystyle X^{\ast }}$ isuniformly convex (and vice versa, via reflexivity).^[3] The moduli of convexity and smoothness are linked by

${\displaystyle {\rho }_{X^{\ast }}(t)=sup\{t\epsilon /2-{\delta }_X(\epsilon ):\epsilon \in [0,2]\},t\ge 0,}$

and the maximal convex function majorated by the modulus of convexity δ_X is given by^[4]

${\displaystyle {\tilde{\delta }}_X(\epsilon )=sup\{\epsilon t/2-{\rho }_{X^{\ast }}(t):t\ge 0\}.}$

Furthermore,^[5]

${\displaystyle {\delta }_X(\epsilon /2)\le {\tilde{\delta }}_X(\epsilon )\le {\delta }_X(\epsilon ),\epsilon \in [0,2].}$

• A Banach space is uniformly smooth if and only if the limit

${\displaystyle \underset{t\to 0}{lim}\frac{\Vert x+ty\Vert -\Vert x\Vert }{t}}$

exists uniformly for all${\displaystyle x,y\in S_X}$ (where${\displaystyle S_X}$ denotes theunit sphere of${\displaystyle X}$).

• When 1 <p < ∞, theL^p-spaces are uniformly smooth (and uniformly convex).

Enflo proved^[6] that the class of Banach spaces that admit an equivalent uniformly convex norm coincides with the class ofsuper-reflexive Banach spaces, introduced byRobert 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. ThePisier renorming theorem^[8] states that a super-reflexive space X admits an equivalent uniformly smooth norm for which the modulus of smoothness ρ_X satisfies, for some constant C and some p > 1

${\displaystyle {\rho }_X(t)\le C{t}^p,t>0.}$

It follows that every super-reflexive spaceY admits an equivalent uniformly convex norm for which themodulus of convexity satisfies, for some constant c > 0 and some positive realq

${\displaystyle {\delta }_Y(\epsilon )\ge c{\epsilon }^q,\epsilon \in [0,2].}$

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.

• Uniformly convex space

• Diestel, Joseph (1984),Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, New York: Springer-Verlag, pp. xii+261,ISBN 0-387-90859-5
• Itô, Kiyosi (1993),Encyclopedic Dictionary of Mathematics, Volume 1, MIT Press,ISBN 0-262-59020-4[1]
• Lindenstrauss, Joram; Tzafriri, Lior (1979),Classical Banach spaces. II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Berlin-New York: Springer-Verlag, pp. x+243,ISBN 3-540-08888-1.

• Banach spaces
• Convex analysis