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July 18th, 2017, 02:20 AM   #1
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Should span be a finite linear combination only?

Wiki says infinite linear combination should be excluded off of the definition of span. So should we answer "Yes" to the following question appearing in UC Davis notes?:

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July 18th, 2017, 02:30 AM   #2
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The answer is indeed yes.
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August 17th, 2017, 04:35 PM   #3
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Note that in "functional analysis", where we have infinite dimensional vector spaces and a notion of limits, so that we can add an infinite number of vectors, we can have a "span" of an infinite number of vectors. Fourier series are an example of that.
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September 26th, 2017, 10:00 PM   #4
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Originally Posted by zzzhhh View Post
Wiki says infinite linear combination should be excluded off of the definition of span. So should we answer "Yes" to the following question appearing in UC Davis notes?:

By itself a vector space doesn't contain any axioms which tell you how to handle limits. Thus, conventionally the span when referring to a vector space only refers to finite linear combinations. This is similar to how one learns how to add finitely many numbers before calculus. The discussion of "adding infintely many" numbers together is postponed until calculus so that this operation can be defined properly.

However, by adding additional structure you may gain the machinery to discuss limits. This typically occurs when first introduced to Banach spaces (complete normed vector spaces). In this case one has a norm and a notion of convergence so once can make sense of an infinite linear combination. This has the obvious definition as the limit of the partial linear combinations. In this case one also uses the terminology "span". However, it is typically very clear what is meant by this usage.
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