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 \( {\displaystyle \|y\|\leq \delta } \) then
‖ x + y ‖ + ‖ x − y ‖ ≤ 2 + ϵ ‖ y ‖ . {\displaystyle \|x+y\|+\|x-y\|\leq 2+\epsilon \|y\|.} {\displaystyle \|x+y\|+\|x-y\|\leq 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]
\( {\displaystyle \rho _{X}(t)=\sup {\Bigl \{}{\frac {\|x+y\|+\|x-y\|}{2}}-1\,:\,\|x\|=1,\;\|y\|=t{\Bigr \}}.} \) \)
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
Every uniformly smooth Banach space is reflexive.[2]
A Banach space X {\displaystyle X} X is uniformly smooth if and only if its continuous dual \( X^{*} \) is uniformly convex (and vice versa, via reflexivity).[3] The moduli of convexity and smoothness are linked by
\( {\displaystyle \rho _{X^{*}}(t)=\sup\{t\varepsilon /2-\delta _{X}(\varepsilon ):\varepsilon \in [0,2]\},\quad t\geq 0,} \)
and the maximal convex function majorated by the modulus of convexity δX is given by[4]
\( {\displaystyle {\tilde {\delta }}_{X}(\varepsilon )=\sup\{\varepsilon t/2-\rho _{X^{*}}(t):t\geq 0\}.} \)
Furthermore,[5]
\( {\displaystyle \delta _{X}(\varepsilon /2)\leq {\tilde {\delta }}_{X}(\varepsilon )\leq \delta _{X}(\varepsilon ),\quad \varepsilon \in [0,2].} \)
A Banach space is uniformly smooth if and only if the limit
l \( {\displaystyle \lim _{t\to 0}{\frac {\|x+ty\|-\|x\|}{t}}} \)
exists uniformly for all \( {\displaystyle x,y\in S_{X}} \) (where \( S_{X} \) denotes the unit sphere of X).
When 1 < p < ∞, the Lp-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 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
\( {\displaystyle \rho _{X}(t)\leq C\,t^{p},\quad t>0.} \)
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
\( {\displaystyle \delta _{Y}(\varepsilon )\geq c\,\varepsilon ^{q},\quad \varepsilon \in [0,2].} \)
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
Uniformly convex space
Notes
see Definition 1.e.1, p. 59 in Lindenstrauss & Tzafriri (1979).
Proposition 1.e.3, p. 61 in Lindenstrauss & Tzafriri (1979).
Proposition 1.e.2, p. 61 in Lindenstrauss & Tzafriri (1979).
Proposition 1.e.6, p. 65 in Lindenstrauss & Tzafriri (1979).
Lemma 1.e.7 and 1.e.8, p. 66 in Lindenstrauss & Tzafriri (1979).
Enflo, Per (1973), "Banach spaces which can be given an equivalent uniformly convex norm", Israel J. Math. 13:281–288.
James, Robert C. (1972), "Super-reflexive Banach spaces", Can. J. Math. 24:896–904.
Pisier, Gilles (1975), "Martingales with values in uniformly convex spaces", Israel J. Math. 20:326–350.
Asplund, Edgar (1967), "Averaged norms", Israel J. Math. 5:227–233.
References
Diestel, Joseph (1984). Sequences and series in Banach spaces. Graduate Texts in Mathematics. 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], 97, Berlin-New York: Springer-Verlag, pp. x+243, ISBN 3-540-08888-1.
Functional analysis (topics – glossary)
Spaces
Hilbert space Banach space Fréchet space topological vector space
Theorems
Hahn–Banach theorem closed graph theorem uniform boundedness principle Kakutani fixed-point theorem Krein–Milman theorem min-max theorem Gelfand–Naimark theorem Banach–Alaoglu theorem
Operators
bounded operator compact operator adjoint operator unitary operator Hilbert–Schmidt operator trace class unbounded operator
Algebras
Banach algebra C*-algebra spectrum of a C*-algebra operator algebra group algebra of a locally compact group von Neumann algebra
Open problems
invariant subspace problem Mahler's conjecture
Applications
Besov space Hardy space spectral theory of ordinary differential equations heat kernel index theorem calculus of variation functional calculus integral operator Jones polynomial topological quantum field theory noncommutative geometry Riemann hypothesis
Advanced topics
locally convex space approximation property balanced set Schwartz space weak topology barrelled space Banach–Mazur distance Tomita–Takesaki theory
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