So given a vector space $V$ of dimension $n$ (say over the real or complex numbers), we can consider its dual space $V^*$. Then we can consider multilinear maps from p copies of $V^*$ and q copies of $V$ into the underlying field. These multilinear maps are the p-q tensors on V. They form a vector space $T^p_qV$, since we can add them and multiply by elements of the underlying field, and still get another element in $T^p_qV$.
Now if we take just the p-0 tensors, we can consider their direct sum, from 0 to infinity, to form the vector space $TV$, which, when we endow with the tensor product forms the tensor algebra.
If anything above is wrong, please correct me.
My quesiton is that now a general element in the tensor algebra will be a sum of tensor of differnt ranks. And so it is ill-defined how many dual vectors the overall object would need to be fed to output an element in the underlying field. And so in general an element in the tensor algebra isn't a well-defined tensor.
I am not sure whether my definition of tensor, or of tensor algebra, or both, is wrong. Please can someone clarify.
(I don't think it matters, but the reason I was going through this was while trying to learn about Clifford Algebras and Spinors and such things.)