Approximation by neural networks is not continuous

Abstract It is shown that in a Banach space X satisfying mild conditions, for its infinite, linearly independent subset G, there is no continuous best approximation map from X to the n-span, span n G . The hypotheses are satisfied when X is an L p -space, 1 span n G is not a subspace of X, it is also shown that there is no continuous map from X to span n G within any positive constant of a best approximation.

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