CONSTRUCTIVE COMPACT LINEAR MAPPINGS

CONSTRUCTIVE COMPACT LINEAR MAPPINGS

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##### Date

1997-07

##### Authors

NGWA, Matthias

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##### Abstract

This dissertation opens with a discussion of the distinction between the classical
and constructive notions of Mathematics. There then follows a description of the three
main varieties of modern constructive mathematics: Bishop's constructive mathematics,
the recursive constructive mathematics of the Russian School of Markov, and Brouwcr's
intuitionistic mathematics. We investigate the relationship between compactness, finite
rank and located kernel for a bounded linear mapping of a normed space into a finitedimensional
space in a constructive setting. One of the main results is that a bounded
linear mapping of a normed space onto a finite-dimensional normed space is
constructively compact if and only if its kernel is located. We proceed with the
investigation of compact operators on a Hilbert space, and compact linear mappings of
a normed space, within the framework of Bishop's constructive mathematics. We
characterise the compactness of a bounded linear mapping of a Hilbert space H into C",
and prove the theorems: Let A and B be compact operators on a Hilbert space H, let
C be an operator on H and let a £ C Then aA is compact, A + B is compact, the
Hilbert adjoint A* is compact; An operator on a Hilbert space has an adjoint if and only
if it is weakly compact. We also look at constructive substitutes for the classically
well-known theorems on compact linear mappings: T is compact if and only if T* is
compact; if S is bounded linear operator and if T is compact, then TS is compact; if S
and T are compact, then S+T is compact.

##### Description

A thesis submitted to the Postgraduate School, Ahmadu
Bello University, Zaria-Nigeria in partial fulfilment of the
requirements for the degree of Master of Science.
(JULY, 1997)

##### Keywords

CONSTRUCTIVE,, COMPACT,, LINEAR,, MAPPINGS.