# Infinite sums in vector spaces

In Linear Algebra we often talk about vector spaces and the defined operations there ( excuse my english, I hope I can get my point across ). But whenever infinite sums come up or the possibility thereof, the professor strictly prohibits them and always mentions that they must be finite.

I currently read additional literature and everywhere you find "remember, XYZ must be finite!".

Wikipedia ( in an article about the basis vector ) says: "The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors."

Still, this doesn't make it very clear to me, what the problem is. In calculus you can talk about them just fine and they have lots of nice properties, too.

What is it about Linear Algebra, that makes infinite sums meaningless?

• What's missing is topology. You can't make sense of infinite sums without the structure required to take a limit. This is exactly what happens in functional analysis, which supplements the vector space structure with a topological structure. – symplectomorphic Dec 3 '16 at 17:15
• @symplectomorphic The topology you mean, is the metric? – ArtificiallyIntelligence Sep 22 '18 at 2:11

A vector space has an operation which allows you to sum two things. Using that you can sum three, by summing two and then adding the result with the remaining one. It also allows you to sum four things, and five, and six: in each case you sum two of them, then add each of the remaining ones until you are done.

If you tried to do this with an infinite sum, you would never be done, and then it just does not work.

"Infinite sums" simply do not exist, neither in vector spaces nor anywhere$^*$: what we study (in analysis) under the name of series are limits of a special form. They behave very similar to sums, and that's a useful intuition, but they are most certainly not sums but limits.

(*) One can define situations with infinitary sums, and they are useful sometimes, sure. But thha is irrelevant in the context of this question.

The main issue is that a simple vector space has no definition of distance. In calculus, we can talk about the convergence or divergence of infinite sums because we have a concept of how close the sum gets to a certain value at infinity. A vector space doesn't give us this ability, so we are unequipped to talk about infinite sums in any meaningful way.

To remedy this, mathematicians have things like metric spaces (or inner product spaces, if you've seen those, which induce a metric), where there is a concept of distance with which we can use to take limits of a sort (see comments).

• We do not really talk of infinite sums, we talk of limitts of sequences of finite sums, and that is a completely different thing. Avoiding mixing those two things is the best way to clarify matters. – Mariano Suárez-Álvarez Dec 3 '16 at 17:13
• Very good point, I get sloppy with terminology but you are correct. Edited to reflect. – ImHereSometimes Dec 3 '16 at 17:15
• What is simple vector space for you? – Arpit Kansal Dec 3 '16 at 17:26
• I meant the type of vector space one would first encounter in linear algebra, one with no extra operation equipped to it other than vector addition and scalar multiplication (ie. no metic or topology, etc.) – ImHereSometimes Dec 3 '16 at 17:27