Algebraic infinite sum notation

olivierd

Jun 23, 2017 2 min read

Algebraic infinite sum notation

On an infinite-dimensional vector space one usually defines a linear combination as a finite sum; for instance, if $K$ is a basis of the $\K$ -vector space $V$ , one may write, for $J$ some finite part of $K$ :

$\pmb v = \sum_{\pmb k \in J} a_{\pmb k} \pmb k$

I find this notation cumbersome, because it seems to make the sum dependent on the arbitrary choice of the finite set $J$ , while in fact it doesn’t, provided $J$ contains all the nonzero values to be added.

For instance, if you wish to add two such linear combinations written with different finite sets $J$ and $J'$ , you cannot do so directly, without first arbitrarily choosing some other finite subset $J''$ of $K$ containing $J \cup J'$ :

$\pmb v + \pmb v' = \sum_{\pmb k \in J} a_{\pmb k} \pmb k + \sum_{\pmb k \in J'} a'_{\pmb k} \pmb k = \sum_{\pmb k \in J''} (a_{\pmb k} + a'_{\pmb k}) \pmb k$

while making excuses for the necessity to define the new $a_{\pmb k}$ ‘s and $a'_{\pmb k}$ ‘s as zero and for the arbitrary choice of $J''$ which doesn’t change the result.

I find it more practical to accept writing sums of an arbitrary collection of objects, such as:

$\pmb v = \sum_{\pmb k \in K} a_{\pmb k} \pmb k$

provided we know in advance that only for a finite number of $\pmb k$ are the values $a_{\pmb k} \pmb k$ nonzero.