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Computable structures and operations on the space of continuous functions

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Melnikov A., Ng K.

Fundamenta Mathematicae

2016

tom 233

nr 2

101-141

We use ideas and machinery of effective algebra to investigate computable structures on the space C[0,1] of continuous functions on the unit interval. We show that (C[0,1],sup) has infinitely many computable structures non-equivalent up to a computable isometry. We also investigate if the usual operations on C[0,1] are necessarily computable in every computable structure on C[0,1]. Among other results, we show that there is a computable structure on C[0,1] which computes + and the scalar multiplication, but does not compute the operation of pointwise multiplication of functions. Another unexpected result is that there exists more than one computable structure making C[0,1] a computable Banach algebra. All our results have implications for the study of the number of computable structures on C[0,1] in various commonly used signatures.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Melnikov A.

1 Ng K.

1 2016

Wyniki wyszukiwania

Computable structures and operations on the space of continuous functions