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".

The computation of the vector product as the determinant of the matrix

$\mathbf{a}\times\mathbf{b}=\det \begin{bmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ \end{bmatrix}$

is a significant abuse of notation as $\mathbf{i},\mathbf{j},\mathbf{k}$ are treated as scalars but are in fact vectors.

With Big O notation, we say that some function f "is" O(g(x)) (given some function g, where x is one of f's parameters). Intuitively this notation groups functions according to their growth respective to some parameter; as such, it would be appropriate to use the set membership notation and say that $f \in O(g(x))$ . However, the usual notation is $f = O (g (x))$ , despite the fact that the implied relationship is not symmetric (which the symbol = usually implies). One reason for this is that, as an extension of the abuse, it is useful to overload relation symbols such as < and ≤, such that, for example, f < O(g(x)) means that f's real growth is less than g(x).