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\inG)?
(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.