How is the tensor exponential map defined over a generic Banach space?

Question

Given some vector $x = (x_1,...,x_d)$ in $\mathbb{R}^d$ we may embed it in the tensor algebra $T(\mathbb{R}):= \prod_{n=0}^\infty (\mathbb{R}^d)^{\otimes n}$ via the tensor exponential map $\exp_\otimes$. Specifically, we have that $$\exp_\otimes(x) := \sum_{n=0}^\infty \frac{x^{\otimes n}}{n!}:= \sum_{n=0}^\infty \frac{1}{n!} \sum_{i_1,...,i_n=1}^d x_{i_1}...x_{i_n} e_{i_1} \otimes ... \otimes e_{i_n},$$ where $\{e_1,...,e_d\}$ denotes the canonical basis of $\mathbb{R}^d$. Now, if we consider a generic Banach space $V$ (possibly infinite dimensional), how is $\exp_\otimes $ defined? This is a fairly common map in the context of Rough Path Theory, but I haven't found any reference where $\exp_\otimes$ is properly defined. Thanks in advance for any insight. Also, a reference would suffice.