Tensor Products and Associative Algebras

[Math Amateur]

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...

I am focused on Section 10.3 The Tensor Algebra ... ...

I need help in order to get a basic understanding of Definition 10.5 in Section 10.3 ...Definition 10.5 plus some preliminary definitions reads as follows:View attachment 5550
View attachment 5551In the above text from Cooperstein, in Definition 10.5, we read the following:

" ... ... An element $$x \in \mathcal{T}(V)$$ is said to be homogeneous of degree $$d$$ if $$x \in \mathcal{T}_d (V)$$ ... ..."My question is as follows:

How can x be such that $$x \in \mathcal{T}(V)$$ and $$x \in \mathcal{T}_d (V)$$ ... does not seem possible to me ... ...

... ... because ... ...

... if $$x \in \mathcal{T}(V)$$ then $$x$$ will have the form of an infinite sequence as in the following:$$x = (x_0, x_1, x_2, \ ... \ ... \ , x_{d-1}, x_d, x_{d+1}, \ ... \ ... \ ... \ ... )
$$where $$x_i \in \mathcal{T}_i (V)$$... ... clearly $$x_d$$ is the $$d$$-th coordinate of $$x$$ and so cannot be equal to $$x$$ ... ..

Can someone please clarify this issue ... clearly I am not understanding this definition ...

Peter

 

[Deveno]

The elements of $\mathcal{T}(V)$ are *finite* sequences (they are functions with finite support in $I$).

The elements that are homogeneous are tensors that are of the same "rank". For example, an element of $\mathcal{T}_3(V)$ might look like:

$v_1 \otimes v_2 \otimes v_3 + w_1 \otimes w_2 \otimes w_3$ where each $v_i, w_i \in V$.

It's analogous to the degree of a polynomial, where "homogeneous" of degree $d$ would mean all but the coefficients of $x^d$ are 0.

Remember the injections $\epsilon_i : \mathcal{T}_i(V) \to \bigoplus\limits_i \mathcal{T}_i(V)$? An element of $\mathcal{T}(V)$ is homogeneous if it is in the image of a single such injection.

Rank 0 tensors = scalars.
Rank 1 tensors = vectors.
Rank 2 tensors = 2-tensors, etc.

We're going to create a "giant algebra" of tensors of all ranks.