Abuse of notation

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In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition (while being unlikely to introduce errors or cause confusion). Abusing notation should be contrasted with "misusing" notation which should be avoided.

Common examples occur when speaking of compound mathematical objects. For example, a topological space consists of a set $T$ and a topology $\mathcal{T}$ , and two topological spaces $(T, \mathcal{T})$ and $(T, \mathcal{T'})$ can be quite different if they have different topologies. Nevertheless, it is common to refer to such a space simply as $T$ when there is no danger of confusion or when it is implicitly clear what topology is being considered. Similarly, one often refers to a group $(G, \star)$ as simply $G$ when the group operation is clear from context.

1 Advantages
2 Disadvantages
3 Examples
4 Quotation
5 See also
6 External links

[edit] Advantages

The new use may achieve clarity in the new area in an unexpected way.

[edit] Disadvantages

The new use may borrow arguments from the old area that do not carry over, creating a false analogy.

[edit] Examples

John Harrison cites "the use of f(x) to represent both application of a function f to an argument x, and the image under f of a subset, x, of f's domain".