The unit ball of the complex P(H-3)

Let H be a two-dimensional complex Hilbert space and P(H-3) the space of 3-homogeneous polynomials on H. We give a characterization of the extreme points of its unit ball, B-P(3H), from which we deduce that the unit sphere of P(H-3) is the disjoint union of the sets of its extreme and smooth points. We also show that an extreme point of B-P(3H) remains extreme as considered as an element of B-L(3H). Finally we make a few remarks about the geometry of the unit ball of the predual of P(H-3) and give a characterization of its smooth points

Infinite dimensional Banach spaces of functions with nonlinear properties

The aim of this paper is to show that there exist infinite dimensional Banach spaces of functions that, except for 0, satisfy properties that apparently should be destroyed by the linear combination of two of them. Three of these spaces are: a Banach space of differentiable functions on R(n) failing the Denjoy-Clarkson property; a Banach space of non Riemann integrable bounded functions, but with antiderivative at each point of an interval; a Banach space of infinitely differentiable functions that vanish at infinity and are not the Fourier transform of any Lebesgue integrable function

Unconditional constants and polynomial inequalities

If P is a polynomial on R of degree at most n, given by P(x) = Sigma(alpha is an element of Nm,vertical bar alpha vertical bar <= n) a(alpha)x(alpha), and P(n)(R(m)) is the space of such polynomials, then we define the polynomial vertical bar P vertical bar by vertical bar P vertical bar(x) = Sigma(alpha is an element of Nm,vertical bar alpha vertical bar <= n) vertical bar a(alpha vertical bar)x(alpha). Now if B subset of R(m) is a convex set, we define the norm parallel to P parallel to(B) := sup{vertical bar(x)vertical bar : x is an element of B} on P(n)(R(m)), and then we investigate the inequality vertical bar vertical bar vertical bar P vertical bar vertical bar vertical bar(B) <= C(B)vertical bar vertical bar vertical bar P vertical bar vertical bar vertical bar(B), providing sharp estimates on C(B) for some specific spaces of polynomials. These C(B)'s happen to be the unconditional constants of the canonical bases of the considered spaces