Sia V {\displaystyle V} un insimul , ( F ; + ; ⋅ ) {\displaystyle (F;+;\cdot )} un corpore , ⊕ : V × V → V {\displaystyle \oplus :V\times V\to V} un operation binari (interior) e ⊙ : F × V → V {\displaystyle \odot :F\times V\to V} un operation binari (exterior). Alora ( V ; ⊕ ; ⊙ ) {\displaystyle (V;\oplus ;\odot )} es un spatio vectorial super le corpore F {\displaystyle F} si e solmente si , pro omne u , v , w ∈ V {\displaystyle u,v,w\in V} e α , β ∈ F {\displaystyle \alpha ,\beta \in F} , pro le operation interior vale
V1: u ⊕ ( v ⊕ w ) = ( u ⊕ v ) ⊕ w {\displaystyle u\oplus (v\oplus w)=(u\oplus v)\oplus w} ,
V2: 0 V ∈ V {\displaystyle 0_{V}\in V} con v ⊕ 0 V = 0 V ⊕ v = v {\displaystyle v\oplus 0_{V}=0_{V}\oplus v=v} ,
V3: − v ∈ V {\displaystyle -v\in V} con v ⊕ ( − v ) = ( − v ) ⊕ v = 0 V {\displaystyle v\oplus (-v)=(-v)\oplus v=0_{V}} ,
V4: v ⊕ u = u ⊕ v {\displaystyle v\oplus u=u\oplus v} , e pro le operation exterior vale
S1: α ⊙ ( u ⊕ v ) = ( α ⊙ u ) ⊕ ( α ⊙ v ) {\displaystyle \alpha \odot (u\oplus v)=(\alpha \odot u)\oplus (\alpha \odot v)} ,
S2: ( α + β ) ⊙ v = ( α ⊙ v ) ⊕ ( β ⊙ v ) {\displaystyle (\alpha +\beta )\odot v=(\alpha \odot v)\oplus (\beta \odot v)} ,
S3: ( α ⋅ β ) ⊙ v = α ⊙ ( β ⊙ v ) {\displaystyle (\alpha \cdot \beta )\odot v=\alpha \odot (\beta \odot v)} ,
S4: 1 ⊙ v = v {\displaystyle 1\odot v=v} con 1 ∈ F {\displaystyle 1\in F} le elemento neutre multiplicative. Le operation interior ⊕ {\displaystyle \oplus } es le addition vectorial , e le operation exterior ⊙ {\displaystyle \odot } es le multiplication scalar .