The supremum norm simplifies the definition of uniformly convergent functions, which is shown in the following proposition:
Let $D$ be a set and let $\mathbb F$ denote the field of real numbers or the field of complex numbers. A sequence of functions $f_n:D\to\mathbb F$ is uniformly convergent to a limit function $f:D\to\mathbb F,$ if and only if $$\lim_{n\to\infty}||f_n-f||_\infty=0,$$ where $||\cdot||_\infty$ denotes the supremum norm of the functions $f_n-f$ on $D.$
Proofs: 1
Proofs: 1
