Computable functional analysis: compact operators on computable banach spaces and computable best approximation

The present thesis deals with computable functional analysis and in this context, especially, with compact operators on computable Banach spaces. For this purpose, the representation based approach to computable analysis (TTE) is used. In the first part, computable Banach spaces with computable Schauder bases are introduced and two representations each are defined for the dual space of a computable Banach space and for the space of compact operators between two computable Banach spaces. Using these representations, computable versions of miscellaneous classical results about compact operators on Banach spaces are formulated and proved. The space of compact operators on computable Banach spaces with computable Schauder bases that satisfy some reasonable additional conditions forms a computable Banach space itself, the composition with bounded linear operators turns out to be computable, and a computable version of the Theorem of Schauder is stated. It turns out that well-behaved dual spaces are closely connected to a well-behaved space of compact operators. In the second part, the problem of best-approximation is studied using methods of (computable) functional analysis. Different computability results about the metric projection onto closed convex subsets and onto finite-dimensional linear subspaces of computable normed spaces are shown. Furthermore, a representation of finite-dimensional linear subspaces via some basis is defined and compared with other representations. It is shown that the basis representation contains the same information as the representation via distance functions enriched by the dimension.