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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenWed, 31 Jan 2024 19:05:02 +0000Projections in a normed linear space and a generalization of the pseudo-inverse
https://resolver.caltech.edu/CaltechTHESIS:02122014-075908297
Authors: {'items': [{'email': 'pje@efgh.com', 'id': 'Erdelsky-P-J', 'name': {'family': 'Erdelsky', 'given': 'Philip John'}, 'show_email': 'YES'}]}
Year: 1969
DOI: 10.7907/8GD7-M707
<p>The concept of a "projection function" in a finite-dimensional real or complex normed linear space H (the function P<sub>M</sub> which carries every element into the closest element of a given subspace M) is set forth and examined.</p>
<p>If dim M = dim H - 1, then P<sub>M</sub> is linear. If P<sub>N</sub> is linear for all k-dimensional subspaces N, where 1 ≤ k < dim M, then P<sub>M</sub> is linear.</p>
<p>The projective bound Q, defined to be the supremum of the operator norm of P<sub>M</sub> for all subspaces, is in the range 1 ≤ Q < 2,
and these limits are the best possible. For norms with Q = 1, P<sub>M</sub> is always linear, and a characterization of those norms is given.</p>
<p>If H also has an inner product (defined independently of the norm), so that a dual norm can be defined, then when P<sub>M</sub> is linear
its adjoint P<sub>M</sub><sup>H</sup> is the projection on (kernel P<sub>M</sub>)<sup>⊥</sup> by the dual norm. The projective bounds of a norm and its dual are equal.</p>
<p>The notion of a pseudo-inverse F<sup>+</sup> 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<sup>+</sup>∥, 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<sup>+</sup>)<sup>+</sup> = F for every F. This condition is also sufficient to prove that we have (F<sup>+</sup>)<sup>H</sup> = (F<sup>H</sup>)<sup>+</sup>, where the latter pseudo-inverse is taken using dual norms.</p>
<p>In all results, the real and complex cases are handled in a completely parallel fashion.</p>
https://thesis.library.caltech.edu/id/eprint/8069