Suppose you have a finite dimensional real Hilbert space $V$ and you form the tensor algebra $$T(V) = \mathbb{R} \oplus V \oplus (V\otimes V) \oplus (V\otimes V \otimes V) \oplus \cdots$$ where the multiplication is given by the tensor product. Now imagine you have a finite set of vectors $S$ which you use to form an ideal $I(S)$. From this, we can get the quotient algebra $Q = T(V)/I(S)$ (see another Wikipedia article).

I am interested in comparing elements of $Q$, ideally using an inner product, if it is possible to define one, or else using a distance, if it is possible to define a norm.

The approach I've taken so far is, given an element $x$ of $T(V)$ is to try to project it onto the orthogonal complement of $I(S)$ (call this $P(x)$) in the assumption that there will be a unique element of the equivalence class of $x$ that lives in this orthogonal complement, and that we can then use the inner product of $T(V)$. However I think this assumption is false. Consider the case where $S =$ {$a\otimes a - a$} for a one dimensional vector space $V$ with basis vector $a$. Then $I(S)$ is the vector space with basis $a - a\otimes a$, $a\otimes a - a\otimes a\otimes a$, $a\otimes a\otimes a - a\otimes a\otimes a\otimes a$, $\ldots$.

Intuitively, I would expect $Q$ in this case to be a two dimensional vector space with basis $1, [a]$. Is this correct? Projecting onto the orthogonal complement seems to give zero however.

Is there a standard way to define a norm or an inner product on a quotient algebra such as this? How can it be computed?

Suppose you have a finite dimensional real Hilbert space $V$ and you form the tensor algebra $$T(V) = \mathbb{R} \oplus V \oplus (V\otimes V) \oplus (V\otimes V \otimes V) \oplus \cdots$$ where the multiplication is given by the tensor product. Now imagine you have a finite set of vectors $S$ which you use to form an ideal $I(S)$. From this, we can get the quotient algebra $Q = T(V)/I(S)$ (see this Wikipedia article).

I am interested in comparing elements of $Q$, ideally using an inner product, if it is possible to define one, or else using a distance, if it is possible to define a norm.

The approach I've taken so far is, given an element $x$ of $T(V)$ is to try to project it onto the orthogonal complement of $I(S)$ (call this $P(x)$) in the assumption that there will be a unique element of the equivalence class of $x$ that lives in this orthogonal complement, and that we can then use the inner product of $T(V)$. However I think this assumption is false. Consider the case where $S =$ {$a\otimes a - a$} for a one dimensional vector space $V$ with basis vector $a$. Then $I(S)$ is the vector space with basis $a - a\otimes a$, $a\otimes a - a\otimes a\otimes a$, $a\otimes a\otimes a - a\otimes a\otimes a\otimes a$, $\ldots$.

Intuitively, I would expect $Q$ in this case to be a two dimensional vector space with basis $1, [a]$. Is this correct? Projecting onto the orthogonal complement seems to give zero however.

Is there a standard way to define a norm or an inner product on a quotient algebra such as this? How can it be computed?