View source for Uniformly smooth space

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

:<math>\|x+y\|+\|x-y\| \le 2 + \epsilon\|y\|.</math>

The '''modulus of smoothness''' of a normed space ''X'' is the function &rho;<sub>''X''</sub> defined for every {{nowrap|''t'' &gt; 0}} by the formula<ref>see Definition&nbsp;1.e.1, p.&nbsp;59 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}.</ref>

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

The triangle inequality yields that {{nowrap|&rho;<sub>''X''</sub>(''t''&thinsp;) &le; ''t''}}.  The normed space ''X'' is uniformly smooth if and only if {{nowrap|&rho;<sub>''X''</sub>(''t''&thinsp;) / ''t''}} tends to 0 as ''t'' tends to&nbsp;0.

==Properties==
* Every uniformly smooth [[Banach space]] is [[reflexive space|reflexive]].<ref>Proposition 1.e.3, p.&nbsp;61 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}.</ref>
* A Banach space <math>X</math> is uniformly smooth if and only if its [[Dual space#Continuous dual space|continuous dual]] <math>X^*</math> is [[uniformly convex space|uniformly convex]] (and vice versa, via reflexivity).<ref>Proposition 1.e.2, p.&nbsp;61 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}.</ref> The moduli of convexity and smoothness are linked by
::<math>\rho_{X^*}(t) = \sup \{ t \varepsilon / 2 - \delta_X(\varepsilon) : \varepsilon \in [0, 2]\}, \quad t \ge 0,</math>
:and the maximal convex function majorated by the modulus of convexity &delta;<sub>''X''</sub> is given by<ref>Proposition 1.e.6, p.&nbsp;65 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}.</ref> 
::<math>\tilde \delta_X(\varepsilon) = \sup \{ \varepsilon t / 2 - \rho_{X^*}(t) : t \ge 0\}.</math>
:Furthermore,<ref>Lemma 1.e.7 and 1.e.8, p.&nbsp;66 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}.</ref>
::<math>\delta_X(\varepsilon / 2) \le \tilde \delta_X(\varepsilon) \le \delta_X(\varepsilon), \quad \varepsilon \in [0, 2].</math>
* A Banach space is uniformly smooth if and only if the limit

::<math>\lim_{t\to 0}\frac{\|x+ty\|-\|x\|}{t}</math>

:exists uniformly for all <math>x, y\in S_X</math> (where <math>S_X</math> denotes the [[unit sphere#Unit balls in normed vector spaces|unit sphere]] of <math>X</math>).
*When {{nowrap| 1 &lt; ''p'' &lt; &infin;}}, the [[Lp space|''L''<sup>''p''</sup>-spaces]] are uniformly smooth (and uniformly convex).

[[Per Enflo|Enflo]] proved<ref>{{citation
| last=Enflo | first=Per | authorlink=Per Enflo
| date=1973
| title=Banach spaces which can be given an equivalent uniformly convex norm
| journal=[[Israel Journal of Mathematics]]
| volume=13
| issue=3–4 | pages=281&ndash;288
| doi=10.1007/BF02762802 | doi-access=}}</ref> 
that the class of Banach spaces that admit an equivalent uniformly convex norm coincides with the class of [[Reflexive space#Super-reflexive space|super-reflexive]] Banach spaces, introduced by [[Robert C. James]].<ref>{{citation
| last=James | first=Robert C. |author-link=Robert C. James
| date=1972
| title=Super-reflexive Banach spaces
| journal=[[Canadian Journal of Mathematics]]
| volume=24
| issue=5 | pages=896&ndash;904
| doi=10.4153/CJM-1972-089-7 | doi-access=free}}</ref>  
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 [[Gilles Pisier|Pisier]] renorming theorem<ref>{{citation
| last=Pisier | first=Gilles
| date=1975
| title=Martingales with values in uniformly convex spaces
| journal=[[Israel Journal of Mathematics]]
| volume=20
| issue=3–4
| pages=326&ndash;350
| doi=10.1007/BF02760337 | doi-access=}}</ref> 
states that a super-reflexive space&nbsp;''X'' admits an equivalent uniformly smooth norm for which the modulus of smoothness &rho;<sub>''X''</sub> satisfies, for some constant&nbsp;''C'' and some&nbsp;{{nowrap|''p'' &gt; 1}}

:<math> \rho_X(t) \le C \, t^p, \quad t > 0.</math>

It follows that every super-reflexive space ''Y'' admits an equivalent uniformly convex norm for which the [[Modulus and characteristic of convexity|modulus of convexity]] satisfies, for some constant&nbsp;{{nowrap|''c'' &gt; 0}} and some positive real ''q''

:<math> \delta_Y(\varepsilon) \ge c \, \varepsilon^q, \quad \varepsilon \in [0, 2].</math>

If a normed space admits two equivalent norms, one uniformly convex and one uniformly smooth, the Asplund averaging technique<ref>{{citation
| last=Asplund | first=Edgar
| date=1967
| title=Averaged norms
| journal=[[Israel Journal of Mathematics]]
| volume=5
| issue=4
| pages=227&ndash;233
| doi=10.1007/BF02771611 | doi-access=}}</ref> produces another equivalent norm that is both uniformly convex and uniformly smooth.

==See also==

* [[Uniformly convex space]]

== Notes ==
{{Reflist}}

==References==

*{{citation
 | last = Diestel 
 | first = Joseph 
 | title = Sequences and series in Banach spaces 
 | series = Graduate Texts in Mathematics 
 | volume = 92 
 | publisher = Springer-Verlag 
 | location = New York 
 | year = 1984 
 | pages = xii+261 
 | isbn = 0-387-90859-5 
 | url = https://archive.org/details/sequencesseriesi0000dies
 | url-access = registration
 }}
*{{citation
|author=Itô, Kiyosi
|title=Encyclopedic Dictionary of Mathematics, Volume 1
|year=1993
|publisher=MIT Press
|isbn=0-262-59020-4
}}  [https://books.google.com/books?id=WHjO9K6xEm4C]
*{{citation
 | last1 = Lindenstrauss
 | first1 = Joram | author1-link = Joram Lindenstrauss
 | last2 = Tzafriri | first2 = Lior
 | title = Classical Banach spaces. II. Function spaces
 | series = Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas]
 | volume = 97 
 | publisher = Springer-Verlag
 | location = Berlin-New York
 | year = 1979
 | pages = x+243
 | isbn = 3-540-08888-1 
}}.

{{Banach spaces}}
{{Functional analysis}}

[[Category:Banach spaces]]
[[Category:Convex analysis]]