On an infinite-dimensional vector space one usually defines a linear combination as a finite sum; for instance, if is a basis of the -vector space , one may write, for some finite part of :
I find this notation cumbersome, because it seems to make the sum dependent on the arbitrary choice of the finite set , while in fact it doesn’t, provided contains all the nonzero values to be added.
For instance, if you wish to add two such linear combinations written with different finite sets and , you cannot do so directly, without first arbitrarily choosing some other finite subset of containing :
while making excuses for the necessity to define the new ‘s and ‘s as zero and for the arbitrary choice of which doesn’t change the result.
I find it more practical to accept writing sums of an arbitrary collection of objects, such as:
provided we know in advance that only for a finite number of are the values nonzero.