Functions are generally studied in normed vector spaces (see the article "Spaces for mathematicians"). Several norms may be available. But unlike in ℝ*n* (and, more generally, in finite-dimensional spaces), norms are not always equivalent: they do not necessarily preserve the same topological properties (convergence, continuity, completion, etc.). Choosing one norm over another is therefore always a delicate matter. Depending on the problem at hand (approximation by polynomial functions, solving partial differential equations, etc.), we choose a norm suited to the technique being used. The norm ||.||2 is often particularly useful, thanks to the inner product underlying it and the tools of Hilbert spaces it makes available.
Convergence... or not ---------------------
Here are a few examples of norms for a function f in the space of continuous functions on the interval [0, 1]:
∣∣f∣∣1=∫01∣f(t)∣dt  ;||f||_1 = \int_0^1 |f(t)|dt\; ;
∣∣f∣∣2=∫01f(t)2dt  ;||f||_2 = \sqrt { \int_0^1 f(t)^2 dt}\; ;