Switching Order of Indices in Summation Notation

[Benny]

Hi, can someone please tell me whether or not I can switch the 'order' of the indices over which a double sum is taken? To clarify, my question is whether or not the following is true.

$$\sum\limits_{j = 1}^n {\sum\limits_{i = 1}^n {\left( {a_i b_j } \right)} } \mathop = \limits^? \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\left( {a_i b_j } \right)} }$$

Any help would be appreciated.

 

[Hurkyl]

Yes, by Fubini's theorem.

Er... I mean by associativity and commutativity! (That's just a finite sum of numbers!)

Or, you could always try proving it by induction!

 

[Astronuc]

Well, one can simply show

a₁(b₁+b₂) + a₂(b₁+b₂) = (a₁+a₂) b₁ + (a₁+a₂) b₂

and the show it for 1,2,3 or 1,2, . . . n

 

[Benny]

Oh ok thanks for the help guys. I was just after a 'yes' or 'no' answer seeing as I assumed this property for one of the questions I was doing.

Seeing as Fubini's theorem was mentioned, is it related to my question in some way that I'm not seeing? I know that Fubini's theorem has something to do with multiple integrals which in turn has something to do with multiple sums but I don't see the connection with my question. It doesn't really matter though because I was working on a question which wasn't related to integrals, well not exactly anyway - it was a vector identity.

Thanks for the help.

 

[Hurkyl]

There is a perspective in which a summation really is just an integral, and this interchange property could be proved with Fubini's theorem in this context.

But there's no reason you would go through such great lengths to prove such an elementary result -- I just like being silly. (And I had a teacher who liked to justify interchanging finite sums with Fubini's theorem)

 

[HallsofIvy]

Notice, by the way, that this is true for finite sums. It is not necessarily true that we can interchange infinite sums.

 

[Benny]

Oh ok, I remember googling something about double sums and there were different representations given for finite and infinite double sums.