Denote $V:=\mathbb{R}^d$ and consider the Cartesian product $V^\infty:=\prod_{k=0}^\infty V^{\otimes k}$ together with its canonical projections $\pi_k : V^\infty\rightarrow V^{\otimes k}, (v_0, v_1, \ldots)\mapsto v_k$, where $V^0:=\mathbb{R}$ and each factor space $V^{\otimes k}$ is equipped with the usual (Euclidean) tensor norm $\|\cdot\|_k$ inherited from $V$.
Define the space
$$\tag{1}E \,:=\,\left\{v\in V^\infty \ \middle| \ \sum_{k\geq 0}\|\pi_k(v)\|_k\cdot\lambda^k<\infty, \ \forall\,\lambda>0\right\}$$
and endow it with the locally convex topology induced by the family of norms $\|v\|_\lambda:=\sum_{k\geq 0}\|\pi_k(v)\|_k\lambda^k$, $\lambda>0$.
For each $v\equiv(v_k)\in V^\infty$ with $v_0=1$, define the (formal) logarithm $\log(v)$ as
$$\tag{2} \log(v)\, := \, \sum_{k\geq 1}\frac{(-1)^{k-1}}{k}(v-1)^{\otimes k}.$$
Question: Is $v\mapsto\log(v)$ continuous as a map from $E_1:=\{v\equiv(v_k)\in E\mid v_0=1\}$ to $E$?
Continuity w.r.t. the product topology on $E$ is trivial (as $\pi_k\circ\log$ is a polynomial for each $k\geq 0$), but the required case seems less clear. Any hints or references are appreciated.
