Approximation of differentiable functions on a Hilbert space. III

C1-fine approximation of functions on Banach spaces with unconditional basis

“…Variations on Moulis's results can be found in [7], although there is a gap in the proof of the generalization of [10, Theorem 2] claimed in [8] and announced in [6]. Indeed, in [6,7] Heble makes a (correct) proof for the C kfine approximation of C k -smooth maps by C ∞ maps on a dense subset D of X , and then he claims to show that in fact D = X , but this last part of the proof is wrong.…”

“…Indeed, in [6,7] Heble makes a (correct) proof for the C kfine approximation of C k -smooth maps by C ∞ maps on a dense subset D of X , and then he claims to show that in fact D = X , but this last part of the proof is wrong. In [8], he claims that he can extend the result in [7] from D to all of X ; this proof is also flawed and it is not clear at all how one could mend it.…”