# What does “norms less than $\delta$” refers to in the context of Riemann sum?

What does "norms less than $\delta$" refers to in the context of Riemann sum?

Given this problem "Prove if $\int_{a}^{b} f(x)$ exists, then for every $\epsilon>0$, there is a $\delta >0$ such that $|\sigma_1 -\sigma_2|<\epsilon$, if $\sigma_1$ and $\sigma_2$ are Riemann sums of $f$ over partitions P$_1$ and P$_2$ of $[a,b]$ with norms less than $\delta$."

I am trying to understand this problem that I posted Prove that if $\int_{a}^{b} f(x)$ exists, $\delta >0$ such that $|\sigma_1 -\sigma_2|<\epsilon$ ), I did receive one answer but I am not understanding the how and the why of his argumentation leads to the conclusion that "the norms is less than $\delta$" (sub-question "norms" why plural?). What should be understood from by "the norms is less than $\delta$"?

• – Namaste Jul 26 '17 at 14:31

The plural concerns the two subdivisions or partitions $P_1$ and $P_2$.
if $P=(x_0,x_1,x_2,...,x_n)$ is a subdivision of $[a,b]$ then what you call norm of $P$ is
$$\|P\|=\max_{0\le i\le n-1}\{x_{i+1}-x_i\}$$
Before this there should be a definition of the "norm of a partition". The norms being referred to here are the norm of $P_1$ and the norm of $P_2$; both are less than $\delta$.