According to the following blog-post, https://unapologetic.wordpress.com/2011/07/06/tensor-bundles/
The vector space $V$ and its dual $V*$ are combined to form the “tensors of type $(r,s)$”:
$\displaystyle T^r_s(V)=V\otimes\dots\otimes V\otimes V^*\otimes\dots\otimes V^*$
I'm curious as to whether there exists a relationship between $T^r_s(V)$ and the tensor algebra $T(V)$ (or $k$-th tensor power $T^k{V}$).
The wiki article https://en.wikipedia.org/wiki/Tensor_algebra defines the tensor algebra as follows:
$T(V)= \bigoplus_{k=0}^\infty T^kV = K\oplus V \oplus (V\otimes V) \oplus (V\otimes V\otimes V) \oplus \cdots.$
Following the previous trends of direct sum, do any relationships like these exist:
$T^k(V)=\bigoplus_{r+s=k}^k T^r_s(V)$
and,
$T(V)=\bigoplus_{r,s=0}^\infty T^r_s(V)$, with $r,s\in\mathbb{N_0}$
Any additional clarity would be greatly appreciated!