normed vector space (original) (raw)

Let 𝔽 be a field which is either ℝ or ℂ. A over 𝔽 is a pair (V,∥⋅∥) where V is a vector space over 𝔽 and ∥⋅∥:V→ℝ is a function such that

1. 1. ∥v∥≥0 for all v∈V and ∥v∥=0 if and only if v=0 in V (positive definiteness)
2. 1. ∥λ⁢v∥=|λ|⁢∥v∥ for all v∈V and all λ∈𝔽

The function ∥⋅∥ is called a norm on V.

Some properties of norms:

1. 1. If W is a subspace of V then W can be made into a normed space by simply restricting the norm on V to W. This is called the induced norm on W.
2. 1. Any normed vector space (V,∥⋅∥) is a metric space under the metric d:V×V→ℝ given by d⁢(u,v)=∥u-v∥. This is called the metric induced by the norm ∥⋅∥.
4. 1. In this metric, the norm defines a continuous map from V to ℝ - this is an easy consequence of the triangle inequality.
5. 1. If (V,⟨,⟩) is an inner product space, then there is a natural induced norm given by ∥v∥=⟨v,v⟩ for all v∈V.