norm
A norm is the abstract notion of “length of a vector.” It strips away everything about an arrow except how far it reaches from the origin, and it does so in a way that respects scaling and triangle geometry. Once you have a norm you can talk about big and small vectors, about how close two vectors are, and about whether a sequence settles down — all the machinery of calculus reappears on objects far stranger than numbers.
Precisely, a norm on a vector space V is a function ||·|| : V -> [0, infinity) with three properties: positive definiteness, ||x|| = 0 if and only if x = 0; absolute homogeneity, ||c x|| = |c| ||x|| for every scalar c and vector x; and subadditivity, ||x + y|| <= ||x|| + ||y|| for all x, y. The last property is the triangle inequality in disguise, and it is what guarantees that d(x, y) = ||x - y|| behaves like a real distance.
If you drop the requirement that ||x|| = 0 forces x = 0 you get a seminorm, which can assign length zero to nonzero vectors; seminorms are useful but do not separate points. A norm is the special, point-separating case, and it is the right amount of structure to build a clean theory of size and approximation.
On the space C[0, 1] of continuous functions the sup norm ||f|| = max over [0, 1] of |f(x)| is a genuine norm: ||f|| = 0 forces f = 0 everywhere, scaling pulls out |c|, and max|f + g| <= max|f| + max|g|. For f(x) = x and g(x) = 1 - x, ||f|| = ||g|| = 1 while ||f + g|| = 1 <= 2, illustrating subadditivity.
The sup norm on continuous functions, with the triangle inequality made concrete.
From the three axioms one derives the reverse triangle inequality | ||x|| - ||y|| | <= ||x - y||, which shows that the norm itself is a continuous function. Continuity of the norm is used constantly, for instance to pass limits inside ||·||.