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Projections in a Normed Linear Space and a Generalization of the Pseudo-Inverse

Citation

Erdelsky, Philip John (1969) Projections in a Normed Linear Space and a Generalization of the Pseudo-Inverse. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/8GD7-M707. https://resolver.caltech.edu/CaltechTHESIS:02122014-075908297

Abstract

The concept of a "projection function" in a finite-dimensional real or complex normed linear space H (the function PM 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 PM is linear. If PN is linear for all k-dimensional subspaces N, where 1 ≤ k < dim M, then PM is linear.

The projective bound Q, defined to be the supremum of the operator norm of PM for all subspaces, is in the range 1 ≤ Q < 2, and these limits are the best possible. For norms with Q = 1, PM 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 PM is linear its adjoint PMH is the projection on (kernel PM) by the dual norm. The projective bounds of a norm and its dual are equal.

The notion of a pseudo-inverse F+ of a linear transformation F is extended to non-Euclidean 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 = (FH)+, where the latter pseudo-inverse 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) ; Pseudo-Inverse, Approximation
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
• Todd, John
Thesis Committee:
• Bohnenblust, Henri Frederic (chair)
• Apostol, Tom M.
• Ryser, Herbert J.
• Todd, John
Defense Date:1 January 1969
Funders:
Funding AgencyGrant Number
Record Number:CaltechTHESIS:02122014-075908297
Persistent URL:https://resolver.caltech.edu/CaltechTHESIS:02122014-075908297
DOI:10.7907/8GD7-M707
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