Citation
Wells, John Campbell (1969) Smooth Banach spaces and approximations. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:01252016133502160
Abstract
If E and F are real Banach spaces let C^{p,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 C^{p,q} smooth if C^{p,q}(E,R) contains a nonzero function with bounded support. This generalizes the standard C^{p} smoothness classification.
If an L^{p} space, p ≥ 1, is C^{q} smooth then it is also C^{q,q} smooth so that in particular L^{p} for p an even integer is C^{∞,∞} smooth and L^{p} for p an odd integer is C^{p1,p1} smooth. In general, however, a C^{p} smooth Bspace need not be C^{p,p} smooth. C_{o} is shown to be a nonC^{2,2} smooth Bspace although it is known to be C^{∞} smooth. It is proved that if E is C^{p,1} smooth then C_{o}(E) is C^{p,1} smooth and if E has an equivalent C^{p} norm then c_{o}(E) has an equivalent C^{p} norm.
Various consequences of C^{p,q} smoothness are studied. If f ϵ C^{p,q}(E,F), if F is C^{p,q} smooth and if E is nonC^{p,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 C^{p,q} smooth if and only if E admits C^{p,q} partitions of unity; E is C^{p,p}smooth, p ˂∞, if and only if every closed subset of E is the zero set of some C^{P} function.
f ϵ C^{q}(E,F), 0 ≤ q ≤ p ≤ ∞, is said to be C_{p,q} approximable on a subset U of E if for any ϵ ˃ 0 there exists a g ϵ C^{p}(E,F) satisfying
sup/xϵU, O≤k≤q ‖ D^{k} f(x)  D^{k} g(x) ‖ ≤ ϵ.
It is shown that if E is separable and C^{p,q} smooth and if f ϵ C^{q}(E,F) is C_{p,q} approximable on some neighborhood of every point of E, then F is C_{p,q} approximable on all of E.
In general it is unknown whether an arbitrary function in C^{1}(l^{2}, R) is C_{2,1} approximable and an example of a function in C^{1}(l^{2}, R) which may not be C_{2,1} approximable is given. A weak form of C_{∞,q}, q≥1, to functions in C^{q}(l^{2}, R) is proved: Let {U_{α}} be a locally finite cover of l^{2} and let {T_{α}} be a corresponding collection of HilbertSchmidt operators on l^{2}. Then for any f ϵ C^{q}(l^{2},F) such that for all α
sup ‖ D^{k}(f(x)g(x))[T_{α}h]‖ ≤ 1.
xϵU_{α},‖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)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  18 November 1968  
Funders: 
 
Record Number:  CaltechTHESIS:01252016133502160  
Persistent URL:  http://resolver.caltech.edu/CaltechTHESIS:01252016133502160  
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  
Last Modified:  25 Apr 2016 23:58 
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