I am so confused in the usage of the words: Map and Mapping.
For example, consider the mapping: $T:W^{2,p}(\Omega)\rightarrow L^{p}(\Omega)$ defined by $T(u):=\Delta u.$ Can we write: consider the map $T$ in this case? When do we use map? and mapping?
Thanks.
While they strictly mean different things, they are often used synonymously by most people, just like we abuse language elsewhere. For example, when we say consider the function $f(x),$ which strictly speaking is not a function, but an expression; more over, since we have not specified the domain or codomain. But this is often used and doesn't cause much confusion.
Similarly here. Although the word map refers to the image of the mapping, while the word mapping is synonymous with function, transformation, and similar words; we often use them interchangeably without real confusion. It reminds me of the use of transformation/transform in a similar cavalier way in the theory of integral transforms. The transformation or mapping is different from its product, the transform or map, but we often identify the function $f:A\to B$ with its image $f(x),$ where $x\in A.$
In other words, the mapping is the operation, whereas the map is the product of the operation. However, just like we usually identify a function and an expression embodying its image, we usually identify a mapping and the map embodying its image, without serious confusion.
To apply this to your example, what would be most appropriately called the map is $T(u),$ whereas $T$ is most properly called the mapping. But few people bother about these distinctions.
Even though I am not a native speaker, I would like to oppose the answer of Allawonder. In differential geometry we have "chart" $h$ from open subset of a manifold to some subset of $\mathbb{R}^n$.
I would call the range of a mapping T "the map" of M rather than $T(m)$ for some particular $m \in M$.
$\mbox{ran}(T)=\{y:\exists x: \langle x,y\rangle\in T\}$
Usual use case is, for example, "the world map", which is given below, certainly is about the range. You can "pick a point on a map", so it is some set, rather than a particular element.
