The question is related to the link https://www.sciencedirect.com/science/article/pii/S0022123619302460#br0110 where in page 12 they define the vector space of formal tensor series over $\mathbb{R}^{d+1}$ by
$T((\mathbb{R^{1+d}}))=\prod_{k=0}^\infty$$(\mathbb{R}^{d+1})^{\otimes k}$
and in the very next line the vector space of polynomials is defined over $\mathbb{R}^{d+1}$ by
$T(\mathbb{R^{1+d}})=\sum_{k=0}^\infty$$(\mathbb{R}^{d+1})^{\otimes k}$
Can anyone please describe me a bit what is difference between this expression? I can see one uses product and other is summation but with that alone I do not understand the significance/difference. As I understand any series is a summation but then how formal tensor series is defined as a product?
I would love to hear any comments.