Uniformly smooth space

In mathematics, a uniformly smooth space is a normed vector space X satisfying the property that for every ϵ>0 there exists δ>0 such that if x,y∈X with ‖x‖=1 and ‖y‖≤δ then

‖x+y‖+‖x−y‖≤2+ϵ‖y‖.

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

ρX(t)=sup⁡{‖x+y‖+‖x−y‖2−1:‖x‖=1,‖y‖=t}.

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\.

## Contents

* 1 Properties
* 2 See also
* 3 Notes
* 4 References

## Properties

* Every uniformly smooth Banach space is reflexive.[2]
* A Banach space 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

ρX*(t)=sup⁡{tε/2−δX(ε):ε∈[0,2]},t≥0,
and the maximal convex function majorated by the modulus of convexity δX is given by[4]

δ~X(ε)=sup⁡{εt/2−ρX*(t):t≥0}.
Furthermore,[5]

δX(ε/2)≤δ~X(ε)≤δX(ε),ε∈[0,2].

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

limt->0‖x+ty‖−‖x‖t

exists uniformly for all x,y∈SX (where SX 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

ρX(t)≤Ctp,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

δY(ε)≥cεq,ε∈[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

1. ↑ see Definition 1.e.1, p. 59 in (Lindenstrauss Tzafriri).
2. ↑ Proposition 1.e.3, p. 61 in (Lindenstrauss Tzafriri).
3. ↑ Proposition 1.e.2, p. 61 in (Lindenstrauss Tzafriri).
4. ↑ Proposition 1.e.6, p. 65 in (Lindenstrauss Tzafriri).
5. ↑ Lemma 1.e.7 and 1.e.8, p. 66 in (Lindenstrauss Tzafriri).
6. ↑ "Banach spaces which can be given an equivalent uniformly convex norm", Israel Journal of Mathematics 13 (3–4): 281-288, 1973, doi:10.1007/BF02762802
7. ↑ "Super-reflexive Banach spaces", Canadian Journal of Mathematics 24 (5): 896-904, 1972, doi:10.4153/CJM-1972-089-7
8. ↑ Pisier, Gilles (1975), "Martingales with values in uniformly convex spaces", Israel Journal of Mathematics 20 (3–4): 326-350, doi:10.1007/BF02760337
9. ↑ Asplund, Edgar (1967), "Averaged norms", Israel Journal of Mathematics 5 (4): 227-233, doi:10.1007/BF02771611

## 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, https://archive.org/details/sequencesseriesi0000dies
* Itô, Kiyosi (1993), Encyclopedic Dictionary of Mathematics, Volume 1, MIT Press, ISBN 0-262-59020-4 [1]
* Lindenstrauss; 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 .

* v
* t
* e

Functional analysis (topics)

Topological vector spaces|

* Asplund
* Banach (list)
* Banach lattice
* Barrelled
* Bornological
* Brauner
* F-space
* Fréchet (tame)
* Hilbert (Inner product space
* Polarization identity)
* LF-space
* Locally convex (Seminorms/Minkowski functionals)
* Mackey
* Montel
* Nuclear
* Normed (norm)
* Quasinormed
* Reflexive
* Riesz
* Smith
* Stereotype
* Strictly convex
* Webbed
* Topological tensor product (of Hilbert spaces)

Topologies of function spaces|

* Dual
* Dual space (Dual norm)
* Operator
* Ultraweak
* Weak (polar
* operator)
* Mackey
* Strong (polar
* operator)
* Ultrastrong
* Uniform convergence

Linear operators|

* Adjoint
* Bilinear (form
* operator
* sesquilinear)
* (Un)Bounded
* Closed
* Compact (on Hilbert spaces)
* (Dis)Continuous
* Densely defined
* Fredholm
* Hilbert–Schmidt
* Functionals (positive)
* Normal
* Nuclear
* Self-adjoint
* Strictly singular
* Trace class
* Transpose
* Unitary

Operator theory|

* Banach algebras
* C*-algebras
* Spectrum (C*-algebra
* radius)
* Spectral theory (of ODEs
* Spectral theorem)
* Polar decomposition
* Singular value decomposition

Theorems|

* Banach–Alaoglu
* Banach–Mazur
* Banach–Saks
* Banach–Schauder (open mapping)
* Banach–Steinhaus (Uniform boundedness)
* Bessel's inequality
* Cauchy–Schwarz inequality
* Closed graph
* Closed range
* Eberlein–Šmulian
* Freudenthal spectral
* Gelfand–Mazur
* Gelfand–Naimark
* Goldstine
* Hahn–Banach (hyperplane separation)
* Kakutani fixed-point
* Krein–Milman
* Lomonosov's invariant subspace
* Mackey–Arens
* Mazur's lemma
* M. Riesz extension
* Riesz representation
* Parseval's identity
* Schauder fixed-point

Analysis|

* Abstract Wiener space
* Bochner space
* Differentiation in Fréchet spaces
* Derivatives (Fréchet
* Gateaux
* functional
* holomorphic)
* Integrals (Bochner
* Dunford
* Gelfand–Pettis
* regulated
* Paley–Wiener
* weak)
* Functional calculus (Borel
* continuous
* holomorphic)
* Inverse function theorem (Nash–Moser theorem)
* Measures (Lebesgue
* Projection-valued
* Vector)
* Weakly measurable function

Types of sets|

* Absolutely convex
* Absorbing
* Balanced
* Bounded
* Convex
* Convex cone (subset)
* Linear cone (subset)
* Radial
* Star-shaped
* Symmetric
* Zonotope

Subsets / set operations|

* Algebraic interior (core)
* Bounding points
* Convex hull
* Extreme point
* Interior
* Minkowski addition
* Polar

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Uniformly smooth space. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Uniformly_smooth_space&oldid=224734"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template