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 $$c\mapsto 0*0*\dots *0 \ $$