16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 4.3.6
Polynomials
Mathematics — Linear Algebra · multiplicative-linear-functionals
Problem
Let F be a field. We have considered certain special linear functionals on F[x] obtained via 'evaluation at t':
L(f)=f(t).
Such functionals are not only linear but also have the property that L(fg)=L(f)L(g). Prove that if L is any linear functional on F[x] such that
L(fg)=L(f)L(g)
for all f and g, then either L=0 or there is a t in F such that L(f)=f(t) for all f.