vector norm

A vector norm is a rule that assigns to every vector a single non-negative number measuring its size. In a first course you met one of them, the Euclidean length ||x||_2 = sqrt(x_1^2 + ... + x_n^2). But that is just one choice. A norm is any function ||.|| that behaves the way a sensible notion of length should.

Precisely, ||.|| must satisfy three axioms. First, positivity: ||x|| >= 0, and ||x|| = 0 only when x is the zero vector. Second, absolute homogeneity: ||c x|| = |c| ||x|| for any scalar c, so scaling a vector scales its length by the same factor. Third, the triangle inequality: ||x + y|| <= ||x|| + ||y||, the rule that no detour is shorter than the straight path. The most common family is the p-norms: ||x||_p = (sum |x_i|^p)^(1/p) for p >= 1, giving the taxicab norm (p=1), the Euclidean norm (p=2), and in the limit the max norm ||x||_inf = max |x_i|.

Why bother with more than one? Because different norms answer different questions. The 1-norm sums error across all entries (total error), the inf-norm reports the single worst entry, and the 2-norm balances them via energy. When you analyze an algorithm or bound an error, choosing the right norm makes the bound natural.

A reassuring fact in finite dimensions: although these norms give different numbers, they are all equivalent — each is bounded above and below by a constant multiple of any other. So a sequence that converges in one norm converges in all of them. The choice changes the constants in your bounds, never the qualitative conclusions.