- #1
Adgorn
- 130
- 18
Homework Statement
Suppose T:V→U is linear and u ∈ U. Prove that u ∈ I am T or that there exists ##\phi## ∈ V* such that TT(##\phi##) = 0 and ##\phi##(u)=1.
Homework Equations
N/A
The Attempt at a Solution
Let ##\phi## ∈ Ker Tt, then Tt(##\phi##)(v)=##\phi##(T(v))=0 ∀T(v) ∈ I am T. So obviously if u ∈ I am T than ##\phi##(u)=0. I now need to prove that if u ∉ I am T, than there exists a linear functional which answers the above criteria, and this is where I'm stuck. I don't know which mapping I define that would answer the criteria.
Any help would be appreciated,