In the definition of the tensor algebra associated with the vector space $V$ over a field $\Bbbk$,
$$
T(V)= \bigoplus_{k=0}^{\infty} V^{\otimes k}
$$
writing it all out we get
$$
T(V)= \Bbbk \oplus V \oplus (V \otimes V) \oplus (V \otimes V \otimes V) \oplus \cdots \oplus (V \otimes \cdots \otimes V) \oplus \cdots.
$$
In this definition, the direct sum is between different objects, namely $\Bbbk \oplus V \oplus (V\otimes V) \oplus \cdots$ etc. However, I cannot find any sources explaining how this would work, could someone help me with this definition?
My sources are https://en.wikipedia.org/wiki/Tensor_algebra and "An Introduction to Clifford Algebras and Spinors by Jayme Vazz, Roldão da Rocha."
Best Answer
Given a family of $k$-vector spaces $\{V_i\}_{i \in I}$, the direct sum $$V := \bigoplus_{i \in I} V_i$$ is defined to be the set of all functions $\alpha : I \to \bigcup_{i \in I} V_i$ such that $\alpha_i := \alpha(i) \in V_i$ for all $i \in I$, and that $\{i \in I : \alpha_i \neq 0\}$ is a finite set. Note that each $V_j$ can be naturally embedded in $V$ by identifying $v_j \in V_j$ with the function $v \in V$ such that $v_i = 0$ for all $i \neq j$. Thus, we write $\alpha \in V$ as a finite sum $\alpha = \sum_{i \in I} \alpha_i$.
Therefore, for example, $1 + v \otimes v + v \otimes v \otimes v \otimes v \in T(V)$ for $v \in V$.