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I am reading vector space from Friedberg. There in the last section they told about infinite dimensional vector space but there is not sufficient contents. Now my question is why can't we define infinite sum? If this is the case then can anyone please tell me the difference between infinite sum in the series in analysis and here? How infinite sum in series is defined and not here?

I know I'm going wrong somewhere, please help me to find it out.

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    $\begingroup$ Infinite sums in analysis are defined as limits of the sequence of finite partial sums. In general there is no limit in vector spaces. $\endgroup$ – Jens Schwaiger Mar 24 at 2:35
  • $\begingroup$ @Jens Schwaiger please elaborate, I cant understand about how can we define infinite sum by limit of a sequence? And also what are the bounds that we can't do in vector spaces? $\endgroup$ – user639336 Mar 24 at 2:38
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    $\begingroup$ @user639336 What you are asking is not at all related to the dimension of vector spaces. $\endgroup$ – amsmath Mar 24 at 2:55
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It's not that one can't define an infinite sum, the issue is that in a space with a binary operation an infinite sum does not automatically make sense. You can't define an infinite sum solely in terms of the finite sum. You need to construct the sequence of partial sums, which then needs to converge.

However, in order to define convergence, you need something like a topology, and we're no longer talking simply about vector spaces anymore: we've moved on to topological vector spaces. So one could arguably say that in a plain vector space, which explicitly isn't given a topology, you can't define an infinite sum.

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  • $\begingroup$ sir you are saying topological vector spaces, are they define infinite as a limit of a sequence or anything else? But sir whenever it's about infinite sum of a series we write something lile a1e1+a2e2+........ doesn't it seems like ordinary binary operation? Secondly I read somewhere we can add if all elements are zero except finitely many. Please help sir about clearing my ideas. $\endgroup$ – user639336 Mar 24 at 3:22
  • $\begingroup$ sir somehow are you want to mean topological vector spaces as functional analysis? $\endgroup$ – user639336 Mar 24 at 3:31
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    $\begingroup$ @user639 If all are zero but finitely many, that will converge in any topology so we don't need an explicit one. Just writing the infinite sum in general doesn't tell you which element of the vector space you're talking about. What if I wrote $1+1+1+\cdots$? This gives the sequence of partial sums $1,2,3,\ldots$, and this doesn't converge. Would you say that that was a silly example? We can't distinguish this from any other example without a topology. In fact you could define a topology where this actually does converge. $\endgroup$ – Matt Samuel Mar 24 at 3:32
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    $\begingroup$ @user Topological vector spaces certainly do occur frequently in functional analysis, but they are also studied outside of that subject. $\endgroup$ – Matt Samuel Mar 24 at 3:33
  • $\begingroup$ sir this means the only essence of topology is laid on infinite dimensional vector space? I mean in finite one the sum is defined, but in the infinite one the infinite sum is'nt. $\endgroup$ – user639336 Mar 24 at 3:40
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In analysis you probably defined infinte sum as follows. Let us say that $a_n$ is some sequance of real numbers. We define partial sums $S_n$ as follows. $$S_1 = a_1 $$ $$S_2 = a_1 + a_2 $$ $$...$$ $$S_n = a_1 + a_2 + ... + a_n$$ Now we define: $$S = \sum_{n=1}^{\infty}{a_n} := \lim_{n \to \infty}{S_n} $$ The point of this is that you see that it is good to have a concept of limit (convergance) to define infinte sum. Limit involves, intuitivley speaking, that one things get closer to another; and that requaries notion of distance. If you have a vector space only, you still do not have a way to mesure length of a vector.

So it would be good if you had some way to mesure length of a vector and you can do that by norm. One way to create a norm on your vector space is to induce it with a inner (scalar) product. Then you can define that sequance of vectors $v_n$ converges to some vector $w$ if sequnace of norms $||v_n||$ of vector converges to norm $||w||$. Then you will be able to define infinte sum of vectors because you have notion of convergance.

I kept it brief, but if you do have any question, feel free to ask.

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  • $\begingroup$ It doesn't actually require a notion of distance. You can define infinite sums in topological vector spaces that are not metrizable, like $\mathbb R^{\mathbb R} $ in the product topology. $\endgroup$ – Matt Samuel Mar 24 at 2:48
  • $\begingroup$ @MattSamuel You can even infinite sums in topological (additive) groups. $\endgroup$ – amsmath Mar 24 at 2:54
  • $\begingroup$ @MattSamuel Thanks for comment. I tried to be pedagogical. However from the question asked I estimate that op is not looking for that kind of answer you propose (altrough it is correct). I estimate that he is probably undergrad in math or someone who just encountered vector spaces and mathematical analysis (and is still not able to have general overview), so I kept my answer informative and simple. If you think that op is looking for some other answer feel free to post your answer. $\endgroup$ – Thom Mar 24 at 2:59
  • $\begingroup$ Sure, I actually already did. $\endgroup$ – Matt Samuel Mar 24 at 3:01
  • $\begingroup$ @MattSamuel Great. $\endgroup$ – Thom Mar 24 at 3:02

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