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Natural norms on symmetric tensor products of normed spaces. (English) Zbl 0961.46013

The basics of a metric theory of symmetric tensor products of normed spaces with applications to n-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 n,s E of a vector space E, a trace duality n,s E, n,s E ' is introduced in a natural way and extended to a dual system n,s E,𝒫( n E) by the definition z,q=q L (z) for z n,s E, q𝒫( n E) and q L being the linear functional q L ( n,s E,)𝒫( n E) associated to q. In section 3 this duality is used to define the metric embedding ε s n,s E ' 1( π s n,s E) ' .

The metric theory of symmetric tensor products starts in section 2. First of all, the existence and uniqueness of a tensor norm π s on n,s E (called the projective s-tensor norm) satisfying the isometry 𝒫( n E,F)= 1( π s n,s E,F) is shown. In particular, π s satisfies ( π s n,s E) ' = 1𝒫( n E). With that tensor norm a natural metric surjection J ˜: ˜ π s n,s E ' 1𝒫 nuc n (E) into the space of nuclear polynomials is constructed. (If E ' has AP, then J ˜ is even a metric isomorphism, which implies 𝒫 nuc n (E) ' = 1𝒫 n (E ' ) in section 4.)

The injective s-norm ε s on n,S E is defined as the restriction of the uniform norm on 𝒫( n E ' ) w.r. to the natural embedding J: n,s E𝒫( n E ' ).

Section 5 is devoted to polarization constants of L p -spaces and the interesting formula c(n,E)=sup{c(n,E/G)GCOFIN(E)} is shown for Banach spaces EMAP. Section 6 concerns the extension of polynomials to ultrapowers.

46B28Spaces of operators; tensor products; approximation properties
46G25(Spaces of) multilinear mappings, polynomials
46M05Tensor products of topological linear spaces
15A69Multilinear algebra, tensor products
46M07Ultraproducts of topological linear spaces