The basics of a metric theory of symmetric tensor products of normed spaces with applications to -homogeneous polynomials is presented in this paper for the first time. This good written presentation contains a lot of background information, and the power of tensor products is demonstrated by reproving several recent results on polynomials. After developing the algebraic theory of symmetric tensor products of a vector space , a trace duality is introduced in a natural way and extended to a dual system by the definition for , and being the linear functional associated to . In section 3 this duality is used to define the metric embedding .
The metric theory of symmetric tensor products starts in section 2. First of all, the existence and uniqueness of a tensor norm on (called the projective -tensor norm) satisfying the isometry is shown. In particular, satisfies . With that tensor norm a natural metric surjection into the space of nuclear polynomials is constructed. (If has AP, then is even a metric isomorphism, which implies in section 4.)
The injective -norm on is defined as the restriction of the uniform norm on w.r. to the natural embedding .
Section 5 is devoted to polarization constants of -spaces and the interesting formula is shown for Banach spaces . Section 6 concerns the extension of polynomials to ultrapowers.