Polynomial algebras and smooth functions in Banach spaces

Let An(X) be the algebra of polynomials on a real Banach space X, which is generated by all continuous polynomials of degree not exceeding n. Let m be the minimal integer such that there is a non-compact m-homogeneous polynomial P∈P(Xm;ℓ1). Then n⩾m implies that the uniform closure of An(X) does not contain all polynomials of degree n+1, and hence the chain of closures An(X)¯, n⩾m is strictly increasing. In the rest of the note we give solutions to three problems concerning the behaviour of smooth functions on Banach spaces posed in the literature. In particular, we construct an example of a uniformly differentiable real valued function f on the unit ball of a certain Banach space X, such that there exists no uniformly differentiable function g on λBX, for any λ>1, which coincides with f in some neighbourhood of the origin.