Example

Let E be a complete normable space considered as a Lie group. Then the vector space $U(E)$ is canonically identified with the vector space TS(E) (Differentiable and Analytic Manifolds, R, 13.2.4). Let $m : E \times E \to E$ be addition on E. Then

is equal to TS(m) (Differentiable and Analytic Manifolds, R, 13.2.4). For $t, t'$ in $U(E) = TS(E)$, the image $t * t'$ of the symmetric tensor product $t \otimes t'$ under $m_*$ is therefore TS(m)($t \otimes t'$). By Algebra, Chapter IV, § 5, no. 6, Proposition 7, this image is just the product $tt'$ in the algebra TS(E). Thus the algebra $U(E)$ is identified with the algebra TS(E).