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The supremum norm of a bounded real- or complex-valued function on a set $X$ isFor a continuous function on a compact space, the extreme value theorem makes this supremum a maximum.

A topological ring is a ring equipped with a topology making addition, additive inversion and multiplication continuous functions. An adic topology is obtained from powers of an ideal.

A topological semigroup is a semigroup with a topology for which multiplication is jointly continuous.

A uniform limit of continuous functions is continuous.

For real continuous functions $f,g$ on a compact interval with $g\ge 0$, some $\alpha$ in the interval satisfies $\int fg=f(\alpha )\int g$. If $\int g>0$, the weighted average lies between the minimum and maximum of $f$, and the intermediate value theorem supplies $\alpha$. If $\int g=0$, the bound $|\int fg|\le \sup |f|\int g$ makes both sides zero. Nonnegativity of the weight is essential to this argument.

The Wiener algebra consists of periodic functions with absolutely summable Fourier coefficients, with norm ${\sum }_k|a_k|$. Multiplication corresponds to discrete convolution of coefficients, so the norm is submultiplicative. Consequently $e^f={\sum }_{n\ge 0}{f}^n/n!$ also belongs to the algebra when $f$ does. Every such Fourier series is uniformly convergent and defines a continuous function.

If $\gamma :[0,1]\to \mathbb{C}\setminus \{0\}$ is continuous and closed, choose a continuous lift $\theta$ such thatThen $\mathrm{wind}(\gamma ,0)=\theta (1)-\theta (0)\in \mathbb{Z}$. This agrees with the contour-integral definition when the path is piecewise smooth.