Formal sums over a vector space

Formal sums over endomorphisms

Let be a vector space over and consider formal sums over the endomorphism ring , denoted .¹ We define the following operations: fcalc

Operations

Summation

The sum of a family in with exists iff the are summable for all , and is given by

Multiplication

The product of a finite list in exists iff for every , the set

is summable and is defined as

Importantly, partitioning a product into existent subproducts and taking the product of those will give the same result, but the converse doesn’t hold: Multiplication of formal sums fails to be associative, instead satisfying partial associativity.

Counterexamples

Consider the Formal delta . Then naïve manipulation would suggest

On the other hand, this triple product exists but contains a nonexistant subproduct

Limits of multivariable formal sums

Let

Then exists iff for every the family is summable, and is given by

See also

• Normal ordered product

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Footnotes

1. 1988. Vertex operator algebras and the Monster, §2.1, pp. 49–50 ↩