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: #m/def/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.

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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