An ideal in C[0,1] is a subring I⊂C[0,1] such that for any f∈I and g∈C[0,1], the product fg is in I.
If I⊂C[0,1] is an ideal such that I=C[0,1], then I is called proper.
A maximal ideal M⊂C[0,1] is a proper ideal such that whenever I is an ideal with M⊂I, in fact M=I. (In other words, M is maximal with respect to the partial order given on ideals of C[0,1] by set inclusion.)
For x∈[0,1], define Mx:={f∈C[0,1]∣f(x)=0}. Note that Mx is an ideal in C[0,1].
Answer the following yes-no questions:
- Is Mx is a maximal ideal of C[0,1] for all x∈[0,1]?
- Does there exist a maximal ideal of C[0,1] that is not of the form Mx for some x∈[0,1]?