The Tensor Product Representation of Polynomials of Weak Type in a DF-Space

Abstract

Let $E$ and $F$ be locally convex spaces over $\mathbf{C}$ and let $P{(}^{n}E;F)$ be the space of all continuous $n$-homogeneous polynomials from $E$ to $F$. We denote by ${⨂}_{n,s,\pi }E$ the $n$-fold symmetric tensor product space of $E$ endowed with the projective topology. Then, it is well known that each polynomial $p\in P{(}^{n}E;F)$ is represented as an element in the space $L({⨂}_{n,s,\pi }E;F)$ of all continuous linear mappings from ${⨂}_{n,s,\pi }E$ to $F$. A polynomial $p\in P{(}^{n}E;F)$ is said to be of weak type if, for every bounded set $B$ of $E$, $p{|}_B$ is weakly continuous on $B$. We denote by $P_w{(}^{n}E;F)$ the space of all $n$-homogeneous polynomials of weak type from $E$ to $F$. In this paper, in case that $E$ is a DF space, we will give the tensor product representation of the space $P_w{(}^{n}E;F)$.