User:J D Bowen/Math735 HW5

7.33b)

Consider the function $ x-c \ $. We have $x-c \in M_c, x-c \notin M_b \ $ since $b-c\neq 0 \ $. Hence, $M_c\neq M_b \ $.

7.33c)

Observe that the function $x\mapsto |x-c|$ is continuous and vanishes at $x=c \ $. Since $x-c \ $ is a polynomial, every function in the ideal $(x-c) \ $ is a polynomial. We have $|x-c|\in M_c, |x-c|\notin (x-c), \implies M_c\neq (x-c) \ $.

7.33c)

Let $(f_1, f_2, \dots, f_n) \ $ be a finite list of generators for $M_c \ $. Consider the function $x\mapsto f_1(x)f_2(x)\dots f_n(x) \ $. Since this function is not a linear combination of generators, it is not in the ideal. But we know it IS in the ideal, since $c\mapsto 0*0*\dots *0 \ $