## The symmetric tensor product of a direct sum of locally convex spaces

### Volume 129 / 1998

Studia Mathematica 129 (1998), 285-295
DOI: 10.4064/sm-129-3-285-295

#### Abstract

An explicit representation of the n-fold symmetric tensor product (equipped with a natural topology τ such as the projective, injective or inductive one) of the finite direct sum of locally convex spaces is presented. The formula for $⨂^n_{τ,s} (F_1⨁ F_2)$ gives a direct proof of a recent result of Díaz and Dineen (and generalizes it to other topologies τ) that the n-fold projective symmetric and the n-fold projective "full" tensor product of a locally convex space E are isomorphic if E is isomorphic to its square $E^2$.