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It seems I'm a bit confused about the concept of span. The definition of span makes complete sense to me (i.e. set of all linear combinations of vectors), but I'm confused how this works in the context of the Small Span Theorem:

Let $f$ be continuous on $\left [a, b \right]$. Given $\epsilon > 0$, there exists a partition P : $x_0 < x_1 < ... < x_n$ of the interval $\left [a, b \right]$ s.t. $f$ is bounded on each closed subinterval $\left [x_{j-1}, x_j \right]$, and s.t. the span of $f$ on each closed subinterval is at most $\epsilon$.

At this point, I'm trying to visualize what the theorem is saying, but the one part stopping me is confusion around how the span could be quantified. For the span of two vectors that fills the whole 2D plane, how could it be "at most" $\epsilon$? Are we actually dealing with the set of all $\epsilon > 0$ instead of just one real value?

(This is less important but if anyone also has insight on the applications for the Small Span Theorem & what makes it important, that would be very interesting to learn once I can better understand the concept!)

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    $\begingroup$ I have not heard of this theorem before but I did some digging and found this: ocw.mit.edu/courses/mathematics/… It seems span in this context is different than it is for vectors. To me, the span of a function on a closed interval is more like range, since it is the maximum value of $f$ minus the minimum value of $f$. $\endgroup$ Commented Jun 26, 2020 at 21:43
  • $\begingroup$ @DreiCleaner Thanks for the link! Interesting that span is used in this way -- but the theorem makes a lot more sense knowing that it's out of the context of vectors. $\endgroup$ Commented Jun 26, 2020 at 21:57
  • $\begingroup$ What was the context of the theorem? If it's a section on Riemann integration, then yeah, it has nothing to do with vector span. It's about controlling the oscillation (max value - min value) of $f$ on each subinterval to ensure the function is integrable. $\endgroup$
    – user156205
    Commented Jun 26, 2020 at 22:05
  • $\begingroup$ @user156205 Yes, it is within a section on Riemann integration. What do you mean by controlling the oscillation? Is this mostly being applied to oscillating functions, or are you just talking about analyzing the extrema? $\endgroup$ Commented Jun 26, 2020 at 22:13
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    $\begingroup$ I just meant analyzing the extrema. $\endgroup$
    – user156205
    Commented Jun 26, 2020 at 22:48

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What I'm understanding by "span" in this context seems to be the range of $f$ on a particular interval of a partition. Basically what the theorem is saying is that for a continuous function, you can always partition it in such a way that the maximum deviation of $f$ within any given partition is at most $\epsilon$

This probably would be used to set up the Riemann Integrability of all continuous functions, as a direct consequence of this theorem would be that LUB - MLB will also be bounded by epsilon for any partition

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