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< p<∞, and G is the set of functions computed by the hidden units of a typical neural network (e.g., Gaussian, Heaviside or hyperbolic tangent). If G is finite and 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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