賦範空間 (Normed space)
- 一般常用的vector space X為real number vector, matrix以及continuous function。
- 而field F為real numbers R or complex numbers C.
Norm
- A norm on a vector space X is a function: ∥∥⋅∥∥:X→R+, that satisfies
∀x,y∈X, and c∈F
- ∥∥x∥∥=0 ⇔ x=0.
- ∥∥cx∥∥=∣∣c∣∣∥∥x∥∥.
- ∥∥x+y∥∥≤∥∥x∥∥+∥∥y∥∥.
Normed space
- A normed vector space (X,∥∥⋅∥∥) is a vector space X with a norm ∥∥⋅∥∥, and it usually denoted the norm on space X by ∥∥⋅∥∥X.
- A Banach space is a complete normed space.
- Complete指的是metric space (X,d)中的every Cauchy sequence in X is converent in X, d(x,y)=∥∥x−y∥∥.