Linear Functional Operations in Vector Spaces and Fields

[daveb]

A linear functional is a function g:V to F where V is a vector space over a field F such that if u and v are elements of V and a is an element of F, then g(u+v) = g(u) + g(v) and
g(au) = ag(u)
Let G be the space of all linear functionals on V. Then if $$\oplus_{1}$$ and $$\otimes_{1}$$ are (repectively) addition and scalar multiplication in V, $$\oplus_{2}$$ and $$\otimes_{2}$$ are (repectively) addition and multiplication in F, and $$\oplus_{3}$$ and $$\otimes_{3}$$ are (repectively) addition and scalar multiplication in G, am I correct in saying that the correct way is to write
g(u$$\oplus_{1}$$v) = g(u) $$\oplus_{3}$$ g(v) and
g(a$$\otimes_{1}$$u) = a$$\otimes_{3}$$g(u)?

(God I hope the latex worked out..it's my first time)

 

[Jimmy Snyder]

A linear functional is a function g:V to F where V is a vector space over a field F such that if u and v are elements of V and a is an element of F, then g(u+v) = g(u) + g(v) and
g(au) = ag(u)
Let G be the space of all linear functionals on V. Then if $$\oplus_{1}$$ and $$\otimes_{1}$$ are (repectively) addition and scalar multiplication in V, $$\oplus_{2}$$ and $$\otimes_{2}$$ are (repectively) addition and multiplication in F, and $$\oplus_{3}$$ and $$\otimes_{3}$$ are (repectively) addition and scalar multiplication in G, am I correct in saying that the correct way is to write
g(u$$\oplus_{1}$$v) = g(u) $$\oplus_{3}$$ g(v) and
g(a$$\otimes_{1}$$u) = a$$\otimes_{3}$$g(u)?

(God I hope the latex worked out..it's my first time)

No, this is not correct. g(u) and g(v) are elements of F. Therefore you should write:
g(u$$\oplus_{1}$$v) = g(u) $$\oplus_{2}$$ g(v)
and
g(a$$\otimes_{1}$$u) = a$$\otimes_{2}$$g(u)

Are the following what you are looking for (where g, h$$\in$$G)?
(g$$\oplus_{3}$$h)(u) = g(u) $$\oplus_{2}$$ h(u)
and
(a$$\otimes_{3}$$g)(u) = a$$\otimes_{2}$$g(u)

 

[daveb]

Right! Forgot about that.