Definition:Mapping/Notation/Warning

Notation for Mapping
The notation:
 * Let $\map f x$ be a mapping (or function) ...

is inaccurate and misleading.

If $f: S \to T$ is a mapping, then $\map f x \in T$ for all $x \in S$.

Thus $\map f x$ is a mapping $\Img f$ is a set of mappings.

The point is that, as used here, $\map f x$ is not a mapping, but it is the image of $x$ under $f$.

Hence we should not talk about:
 * the function $\cos x$

but instead should say:
 * the function $\cos$

or:
 * the function $x \mapsto \cos x$

although for the latter it would be better to also specify the domain and codomain.

There may be places on where this point is not rigorously followed.

Work is in progress to correct it.