Strictly convex or “non-singular” continuous homogeneous polynomials in real or complex Banach spaces

Abstract

Fix an even integer d≥2. Let X be a separable real Banach space. Here we prove the existence a closed subspace Y of X and homogeneous degree d polynomials Q, Q ₁, respectively on X and on Y with the following properties:

1. Q(x) ≥ 0 for all x∈X and Q(x)=0 if and only if x∈Y;

2. Q ₁(y)>0 for all y∈Y;

3. Q ₁ is strictly convex, i.e. for every ε > 0 the set B(Q ₁,ε):= {y∈Y: Q ₁(y)≤ε} is convex and Q ₁(tu+(1-t)v))<ε for all u, v∈B(Q ₁,ε) and all 0 <t <1;

4. the continuous homogeneous form Q˜ induced by Q on X / Y is strictly convex.

In the complex case we prove a similer existence result for complete intersections of the projects spaces P(Y) and P(X/Y).

Keywords:

• Banach analytic manifold
• continuous homogeneous form
• analytic function in infinite-dimensional Banach spaces
• infinite-dimensional projective space

AMS Subject Classifications:

• 32K05
• 32C05
• 32D20
• 46E99