The inforation in this question comes from two Wikipedia articles, namely, the articles Tensor Algebra and Exterior Algebra. I'll summarize all of the relevant information.
Let $V$ be an $F$-vector space. Then, if $k\geq 1$ define $T^k V$ to be the vector space of $k$-tensors on $V$, and define $T^0V = F$. Then the tensor algebra $T(V)$ on $V$ is defined as the direct sum of all $T^k V$ for $k\geq 0$, along with the tensor product.
Let $I$ be the ideal in $T(V)$ generated by all elements of $T(V)$ of the form $x\otimes x$ where $x\in V$. Then the exterior algebra on $V$ is defined as $\Lambda(V) = T(V)/I$.
I have two questions about the above.
First, it's my understanding that given two vector spaces $V$ and $W$, the direct sum $V\oplus W$ consists of elements of $V\times W$ along with the obvious vector space structure. The direct sum of vector spaces $V_1, V_2,...$ is given by the subset of $V_1\times V_2\times...$ consisting of elements with only finitely many nonzero components. However, intuitively, it is also my understanding that $T(V)$ should consist of tensors over $V$, but it appears that it consists of $\omega$-tuples of tensors over $V$ (with only finitely many nonzero). What should I make of this discrepancy?
Second, how can we take the tensor product of a vector $x\in V$ with itself in order to construct the ideal $I\subseteq T(V)$? It was my understanding that the tensor product would takes tensors as arguments, and I can't see any natural identification between elements of $V$ and certain elements of $T(V)$.
I apologize if I'm missing something obvious. Thanks in advance.