Chapter 3 Normed Spaces and Their Topology (part I)
We’d like to get to operations like differentiation and integration. We could jump into that right away, and just talk about functions which take in one real number and spit out another real number; that is, we could stick to single-variable calculus. But it turns out that almost all of the ideas of single-variable calculus make sense in much more generality, with not a whole lot more work.
So let’s see what we need to be able to do calculus.
The Four-Step of Calculus.
We understand linear change in a very visceral way. Consider how easy these problems are to answer:
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If I drive for 3 hours at 70 miles per hour, how far did I travel?
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16 baskets of rocks total 64 kilograms. How many kilgrams of rocks does each basket hold?
But, of course, nobody actually drives at a constant speed for 3 hours. And it’s unlikely that each basket has exactly the same amount of rock.
The computations we do in calculus are linearizations. That is, we dance this four-step:
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Pretend that the (nonlinear) phenomenon you want to understand is linear.
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Understand the linear version really well.
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Cross your fingers that the linear model is pretty close to the nonlinear reality.
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Translate the linear understanding back into nonlinear terms.
For example, consider the problem of determining when a function is increasing or decreasing. The steps in that case work like this. To see if \(f(x)\) is increasing at \(x=a\text{:}\)
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Compute \(f'(a)\text{.}\)
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A linear function is increasing if its slope is positive, decreasing if its slope is negative, and constant if its slope is zero.
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???
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The criterion is: if \(f'(a)\gt 0\text{,}\) then \(f\) is increasing near \(x=a\text{;}\) if \(f'(a)\lt 0\text{,}\) then \(f\) is decreasing near \(x=a\text{.}\)
I’ve listed ??? for step 3 because, well, in a typical calculus course you just sort of cross your fingers and hope. Notice something else, too: the slope-zero clause doesn’t carry through to the nonlinear case.
If you think about most things you learned in calculus class, you can probably fit them into this pretend-it’s-linear-and-hope framework. This includes everything from the Chain Rule to applying integration to compute probabilities.
What does linear mean?
In order to pretend something is linear, of course, you have to know what linear means. Linearity is about adding and scaling. So, just as we did when we considered the queston What is a number? to develop the definition of an ordered field, we’ll ask What kind of thing lets us talk about linearity?. That’s what motivates the definition of a normed space.
