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

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 ${\displaystyle {⨂}^n\_\{\tau ,s\}(F_1⨁F_2)}$ 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 ${\displaystyle E^2}$.