Approximation by neural networks is not continuous

It is shown that in a Banach space X satisfying mild conditions, for its in finite, linearly independent subset G, there is no continuous best approxim ation map from X to the n-span, span(n) G. The hypotheses are satisfied whe n X is an L (p)-space, 1 < p < infinity, and G is the set of functions comp uted by the hidden units of a typical neural network (e.g., Gaussian, Heavi side 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. (C) 1999 Elsevier Sc ience B.V. All rights reserved.