Metric spaces impose no requirements on the structure of the set MM. We will
now consider more structured MM, beginning by generalizing the familiar
concept of a vector.
Let KK be a field of scalars, i.e., K=RK=R or CC. Let VV be a set of vectors equipped with two binary operations:
- vector addition: +:V×V→V+:V×V→V
- scalar multiplication: ·:K×V→V·:K×V→V
We say that VV is a vector space (or linear space) over KK if
- VS1: VV forms a group under addition, i.e.,
- (x+y)+z=x+(y+z)(x+y)+z=x+(y+z) (associativity)
- x+y=y+xx+y=y+x (commutativity)
- ∃0∈V∃0∈V such that ∀x∈V∀x∈V, x+0=0+x=xx+0=0+x=x
- ∀x∈V∀x∈V, ∃y∃y such that x+y=0x+y=0
- VS2: For any α,β∈Kα,β∈K and x,y∈Vx,y∈V
- α(βx)=(αβ)xα(βx)=(αβ)x (compatibility)
- (α+β)(x+y)=αx+αy+βx+βy(α+β)(x+y)=αx+αy+βx+βy (distributivity)
- ∃1∈K∃1∈K such that 1x=x1x=x
- RNRN over RR (not RNRN over CC)
- CNCN over CC or CNCN over RR
- Set of polynomials of degree NN with rational coefficients over QQ
- The set of all infinitely-long sequences of real numbers over RR
- GF(2)N:{0,1}NGF(2)N:{0,1}N over {0,1}{0,1} with mod 2 arithmetic (Galois field)
- C[a,b]C[a,b] over RR
