Answer
The norm of a partition in the context of a closed interval \([a, b]\) is a measure of how finely the interval is divided.
Work Step by Step
The norm of a partition in the context of a closed interval \([a, b]\) is a measure of how finely the interval is divided.
Given a partition \(P = \{x_0, x_1, \ldots, x_n\}\) of \([a, b]\), where \(a = x_0 < x_1 < \ldots < x_n = b\), the norm of the partition is defined as follows;
\[ \text{norm}(P) = \max_{i=1}^n (x_i - x_{i-1}) \]
We can also say, it's the length of the longest subinterval in the partition. The smaller the norm, the finer the partition.
For a given function and partition, the Riemann sum depends on the choice of sample points within each subinterval. As the norm of the partition approaches zero (i.e., as the partition becomes finer), the Riemann sum converges to the definite integral over \([a, b]\). This concept is fundamental to the definition of the definite integral in calculus.
