Let V be a vector space over R. A norm on V is a function ∥⋅∥:V→R satisfying the following properties:
- The norm is nonnegative: ∥v∥≥0 for all v∈V, with equality if and only if v=0.
- The norm scales with vectors: ∥cv∥=∣c∣⋅∥v∥ for all v∈V and c∈R.
- The norm satisfies the triangle inequality: ∥v+w∥≤∥v∥+∥w∥ for all v,w∈V.
A vector space equipped with a norm is called, unsurprisingly, a normed vector space.
Suppose V is a normed vector space. For v,w∈V, define a function d:V×V→R by d(v,w)=∥v−w∥. One can verify that this function d is a metric on V, giving V the structure of a metric space.
If the metric space structure on V induced by the norm is complete (i.e., Cauchy sequences converge), then V is called a Banach space.
Consider the space ℓ∞(R), consisting of bounded sequences of real numbers. This is a vector space over R, using coordinate-wise addition and scalar multiplication. One may define a norm on ℓ∞(R) by setting ∥(a1,a2,a3,⋯)∥=i≥1sup∣ai∣. Under this norm, is ℓ∞(R) a Banach space?