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Best Approximation Pair of Two Skew Lines via an ANDERSON-DUFFIN Formula

Vicente, M. A. ; Costa, C. ; Martins, F. ; Beites, P. ; Vitória, José

East Asian Journal on Applied Mathematics Vol. 109, Nº 1, pp. 89 - 104, December, 2018.

ISSN (print): 2079-7362
ISSN (online): 2079-7370

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This is a paper on geometry in the usual real Euclidean space. We
treat the distance between two skew lines, by using a very beautiful
instance of the parallel sum of two matrices. We display the two
points where the shortest distance between two skew lines is achieved.
This paper offers the reader a great amount of facts aiming to
rise awareness about delicate questions - when dealing with infinite
dimensional spaces: on closed sum of subspaces; on the non-linearity
of the projector operator; and on the necessity of the sum of (closed)
subspaces to be closed in order to guarantee the existence of the
Moore-Penrose generalized inverse of the sum of two projectors. In
making use of approximation theory results, this text is sprinkled
with observations regarding concepts stemming from linear algebra to
functional analysis to topology.