CaltechTHESIS
A Caltech Library Service

# Smooth Banach spaces and approximations

## Citation

Wells, John Campbell (1969) Smooth Banach spaces and approximations. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:01252016-133502160

## Abstract

If E and F are real Banach spaces let Cp,q(E, F) O ≤ q ≤ p ≤ ∞, denote those maps from E to F which have p continuous Frechet derivatives of which the first q derivatives are bounded. A Banach space E is defined to be Cp,q smooth if Cp,q(E,R) contains a nonzero function with bounded support. This generalizes the standard Cp smoothness classification.

If an Lp space, p ≥ 1, is Cq smooth then it is also Cq,q smooth so that in particular Lp for p an even integer is C∞,∞ smooth and Lp for p an odd integer is Cp-1,p-1 smooth. In general, however, a Cp smooth B-space need not be Cp,p smooth. Co is shown to be a non-C2,2 smooth B-space although it is known to be C smooth. It is proved that if E is Cp,1 smooth then Co(E) is Cp,1 smooth and if E has an equivalent Cp norm then co(E) has an equivalent Cp norm.

Various consequences of Cp,q smoothness are studied. If f ϵ Cp,q(E,F), if F is Cp,q smooth and if E is non-Cp,q smooth, then the image under f of the boundary of any bounded open subset U of E is dense in the image of U. If E is separable then E is Cp,q smooth if and only if E admits Cp,q partitions of unity; E is Cp,psmooth, p ˂∞, if and only if every closed subset of E is the zero set of some CP function.

f ϵ Cq(E,F), 0 ≤ q ≤ p ≤ ∞, is said to be Cp,q approximable on a subset U of E if for any ϵ ˃ 0 there exists a g ϵ Cp(E,F) satisfying

sup/xϵU, O≤k≤q ‖ Dk f(x) - Dk g(x) ‖ ≤ ϵ.

It is shown that if E is separable and Cp,q smooth and if f ϵ Cq(E,F) is Cp,q approximable on some neighborhood of every point of E, then F is Cp,q approximable on all of E.

In general it is unknown whether an arbitrary function in C1(l2, R) is C2,1 approximable and an example of a function in C1(l2, R) which may not be C2,1 approximable is given. A weak form of C∞,q, q≥1, to functions in Cq(l2, R) is proved: Let {Uα} be a locally finite cover of l2 and let {Tα} be a corresponding collection of Hilbert-Schmidt operators on l2. Then for any f ϵ Cq(l2,F) such that for all α

sup ‖ Dk(f(x)-g(x))[Tαh]‖ ≤ 1.

xϵUα,‖h‖≤1, 0≤k≤q

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Mathematics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
• DePrima, Charles R.
Thesis Committee:
• Unknown, Unknown
Defense Date:18 November 1968
Funders:
Funding AgencyGrant Number
NSFUNSPECIFIED
Record Number:CaltechTHESIS:01252016-133502160
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:01252016-133502160
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9545
Collection:CaltechTHESIS
Deposited By: Leslie Granillo
Deposited On:25 Jan 2016 23:49  Preview