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In Infinite dimensional Clifford algebras?, the answers spoke on how to construct a Clifford algebra in infinite dimensions. What I want to know is: what does $Cl^{p,q,r}(\Bbb R)$ look like when any combination of $p,q,$ and $r$ are infinite. My initial thought was that, seeing as Clifford algebras are based upon quadratic forms, you’d just need 3 elements of $l^2$ as coefficients for the positive, negative, and zero squaring generators.

Later on however, I thought about using a direct limit with the morphisms being the natural inclusion mappings. I, however, do not have familiarity in this field and can’t say if the two constructions are the same or if any one of them could be called natural.

Lastly, to consider a given construction natural, the given space should be complete with respect to the metric generated by the inner product $\langle A,B \rangle\equiv \langle A R(B) \rangle_0$ Where $R(B)$ is just the reverse of $B$. Or maybe it ought to be complete with respect to a more natural metric?

So what is the character of a Clifford algebra in terms of its basis?

Lave Cave
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  • There are several different variations depending on what you want to do. The "algebraic" construction can be expressed as a direct limit as you say, but it won't be complete; so one can contemplate various completions of it. For example there is a $C^{\ast}$-completion. – Qiaochu Yuan Sep 08 '22 at 00:34

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