Citation
Erdelsky, Philip John (1969) Projections in a Normed Linear Space and a Generalization of the PseudoInverse. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/8GD7M707. https://resolver.caltech.edu/CaltechTHESIS:02122014075908297
Abstract
The concept of a "projection function" in a finitedimensional real or complex normed linear space H (the function P_{M} which carries every element into the closest element of a given subspace M) is set forth and examined.
If dim M = dim H  1, then P_{M} is linear. If P_{N} is linear for all kdimensional subspaces N, where 1 ≤ k < dim M, then P_{M} is linear.
The projective bound Q, defined to be the supremum of the operator norm of P_{M} for all subspaces, is in the range 1 ≤ Q < 2, and these limits are the best possible. For norms with Q = 1, P_{M} is always linear, and a characterization of those norms is given.
If H also has an inner product (defined independently of the norm), so that a dual norm can be defined, then when P_{M} is linear its adjoint P_{M}^{H} is the projection on (kernel P_{M})^{⊥} by the dual norm. The projective bounds of a norm and its dual are equal.
The notion of a pseudoinverse F^{+} of a linear transformation F is extended to nonEuclidean norms. The distance from F to the set of linear transformations G of lower rank (in the sense of the operator norm ∥F  G∥) is c/∥F^{+}∥, where c = 1 if the range of F fills its space, and 1 ≤ c < Q otherwise. The norms on both domain and range spaces have Q = 1 if and only if (F^{+})^{+} = F for every F. This condition is also sufficient to prove that we have (F^{+})^{H} = (F^{H})^{+}, where the latter pseudoinverse is taken using dual norms.
In all results, the real and complex cases are handled in a completely parallel fashion.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  (Mathematics) ; PseudoInverse, Approximation  
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:  1 January 1969  
Funders: 
 
Record Number:  CaltechTHESIS:02122014075908297  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:02122014075908297  
DOI:  10.7907/8GD7M707  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  8069  
Collection:  CaltechTHESIS  
Deposited By:  Benjamin Perez  
Deposited On:  12 Feb 2014 17:10  
Last Modified:  26 Apr 2024 23:13 
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