POLYNOMIAL ALGEBRAS AND SMOOTH FUNCTIONS IN BANACH SPACES

According to the fundamental Stone-Weierstrass theorem, if X is a finite dimensional real Banach space, then every continuous function on the unit ball B_X can be uniformly approximated by polynomials. For infinite dimensional Banach spaces the statement of the Stone-Weierstrass Theorem is false, even if we replace continuous functions by the uniformly continuous ones (which is a natural condition that coincides with continuity in the finite dimensional setting): in fact, on every infinite-dimensional Banach space X there exists a uniformly continuous real function not approximable by continuous polynomials. The natural problem of the proper generalization of the result for infinite dimensional spaces was posed by Shilov (in the case of a Hilbert space). Aron observed that the uniform closure on B_X of the space of all polynomials of the finite type is precisely the space of all functions which are weakly uniformly continuous on B_X. Since there exist infinite dimensional Banach spaces such that all bounded polynomials are weakly uniformly continuous on B_X (e.g. C_0 or more generally all Banach spaces not containing a copy of l_1 and such that all bounded polynomials are weakly sequentially continuous on B_X), this result gives a very satisfactory solution to the problem. Unfortunately, most Banach spaces, including L_p, do not have this special property. In this case, no characterization of the uniform limits of polynomials is known. But the problem has a more subtle formulation as well. Let us consider the algebras consisting of all polynomials which can be generated by finitely many algebraic operations of addition and multiplication, starting from polynomials on X of degree not exceeding n. Of course, such polynomials can have arbitrarily high degree. It is clear that, if n is the lowest degree such that there exists a polynomial P which is not weakly uniformly continuous, then the we have equalities among the algebras up to n-1 and then we have a strict inclusion. The problem of what happens from n on has been studied in several papers. The natural conjecture appears to be that once the chain of eualities has been broken, it is going to be broken at each subsequent step. The proof of this latter statement given by Hajek in 1996, for all classical Banach spaces, based on the theory of algebraic bases, is unfortunately not entirely correct, as was pointed out by our colleague Michal Johanis. It is not clear to us if the theory of algebraic bases developed therein can be salvaged. Fortunately, the main statement of this theory can be proved using another approach. The complete proof can be found in this thesis. Most of the results in this area are therefore safe. The main result of this thesis implies all previously known results in this area (all confirming the above conjecture) as special cases. We also give solutions to three other problems posed in the literature, which are concerning smooth functions rather than polynomials, but which belong to the same field of study of smooth mappings on a Banach space. The first result is a construction of a non-equivalent C^k-smooth norm on every Banach space admitting a C^k-smooth norm, answering a problem posed in several places in the literature. We solve a another question by proving that a real Banach space admitting a separating real analytic function whose holomorphic extension is Lipschitz in some strip around X admits a separating polynomial. Eventually, we solve a problem posed by Benyamini and Lindenstrauss, concerning the extensions of uniformly differentiable functions from the unit ball into a larger set, preserving the values in some neighbourhood of the origin. More precisely, 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 cB_X for any c>1 which coincides with f in some neighbourhood of the origin. To do so, we construct suitable renormings of c_0, based on the theory of W-spaces.