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Just finished a course in linear algebra, where the norm of a vector essentially was described as the length of the vector. In calculus, we just started talking about the definite integral of a function, where the norm of a partition came up, being defined as the max size of a subinterval given a set of subintervals.

Are these two uses of norm related, and if so how?

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2 Answers 2

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The max size of the absolute value of a finite list of numbers is the infinity-norm of that list. This is an example of a p-norm. For $p=2$ you get the Euclidean norm that you're familiar with.

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In addition to what @vadim said, it is just used because of norm-like properties such as $$ \|\mathcal{P}\| > 0 $$ for any partition $\mathcal{P}$, and $$ \|\mathcal{P}_1\cup\mathcal{P}_2\| \leq \max\{\|\mathcal{P}_1\|, \|\mathcal{P}_2\|\} $$ In other words, it is some measure of the size of a partition that behaves well with respect to the algebraic operations that one can perform on partitions.

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