normed vector space
Imagine you want to do geometry not with arrows in the plane but with whole functions, or infinite sequences, as your “points.” To get started you only need one thing: a way to measure how long each such object is. A normed vector space is exactly a vector space carrying that single piece of equipment — a norm, written ||x||, that assigns a nonnegative length to every vector in a consistent way.
Formally, a normed vector space is a vector space V over the real or complex numbers together with a function ||·|| : V -> [0, infinity) satisfying three axioms: ||x|| = 0 only when x = 0; ||c x|| = |c| ||x|| for every scalar c (homogeneity); and ||x + y|| <= ||x|| + ||y|| (the triangle inequality). From the norm one defines a distance d(x, y) = ||x - y||, so every normed space is automatically a metric space, and notions like convergence and continuity come for free.
Two cautions. First, the norm is genuine extra structure: the same underlying vector space can carry many inequivalent norms, and in infinite dimensions they need not produce the same convergent sequences. Second, a normed space need not be complete — Cauchy sequences may fail to converge. A normed space that is complete is given the special name Banach space, and completeness is what makes most of analysis work.
On R^2 the Euclidean norm is ||(x, y)|| = sqrt(x^2 + y^2), but ||(x, y)||_1 = |x| + |y| and ||(x, y)||_inf = max(|x|, |y|) are also norms. Their unit balls are a disk, a diamond, and a square — different shapes, yet in two dimensions they are all equivalent.
Three norms, three unit-ball shapes, one equivalence class in finite dimensions.
In finite dimensions all norms are equivalent, so the topology and the notion of convergence do not depend on the chosen norm. This convenient fact fails badly in infinite dimensions, which is precisely why functional analysis is harder and more interesting than linear algebra.