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July 18th, 2017, 02:20 AM  #1 
Newbie Joined: Jun 2017 From: Earth Posts: 17 Thanks: 0  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?: 
July 18th, 2017, 02:30 AM  #2 
Senior Member Joined: Oct 2009 Posts: 439 Thanks: 147 
The answer is indeed yes.

August 17th, 2017, 04:35 PM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 
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.

September 26th, 2017, 10:00 PM  #4  
Senior Member Joined: Sep 2016 From: USA Posts: 435 Thanks: 247 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
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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combination, finite, linear, span 
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