The supremum norm simplifies the definition of uniformly convergent functions, which is shown in the following proposition:

Proposition: Supremum Norm and Uniform Convergence

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

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
  2. Heuser Harro: "Lehrbuch der Analysis, Teil 1", B.G. Teubner Stuttgart, 1994, 11th Edition