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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 $〈{⨁}^{n,s}E,{⨁}^{n,s}{E}^{\text{'}}〉$ is introduced in a natural way and extended to a dual system $〈{⨁}^{n,s}E,𝒫{\left(}^{n}E\right)〉$ by the definition $〈z,q〉={q}^{L}\left(z\right)$ for $z\in {⨁}^{n,s}E$, $q\in 𝒫{\left(}^{n}E\right)$ and ${q}^{L}$ being the linear functional ${q}^{L}\in ℒ\left({⨁}^{n,s}E,ℂ\right)\cong 𝒫{\left(}^{n}E\right)$ associated to $q$. In section 3 this duality is used to define the metric embedding ${⨁}_{{\epsilon }_{s}}^{n,s}{E}^{\text{'}}\stackrel{1}{↪}{\left({⨁}_{{\pi }_{s}}^{n,s}E\right)}^{\text{'}}$.

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

The injective $s$-norm ${\epsilon }_{s}$ on ${⨁}^{n,S}E$ is defined as the restriction of the uniform norm on $𝒫{\left(}^{n}{E}^{\text{'}}\right)$ w.r. to the natural embedding $J:{⨁}^{n,s}E↪𝒫{\left(}^{n}{E}^{\text{'}}\right)$.

Section 5 is devoted to polarization constants of ${L}_{p}$-spaces and the interesting formula $c\left(n,E\right)=sup\left\{c\left(n,E/G\right)\mid G\in \text{COFIN}\left(E\right)\right\}$ is shown for Banach spaces $E\in \text{MAP}$. Section 6 concerns the extension of polynomials to ultrapowers.

##### MSC:
 46B28 Spaces of operators; tensor products; approximation properties 46G25 (Spaces of) multilinear mappings, polynomials 46M05 Tensor products of topological linear spaces 15A69 Multilinear algebra, tensor products 46M07 Ultraproducts of topological linear spaces