Use the Stone-Weierstrass Theorem to describe the subalgebraA ofC[a,b],(ab), generated by {1,x}. Also findAin(C[a,b],).

Definition 2.41 (Algebra/Subalgebra). An algebra over a scalar field F (F = R or F = C) is a vector space A over IF (see Definition 3.1) with an associative and bilinear multiplication of elements ABX,YHIYEA: 1. (xy)Z = X( )Z), X,V.ZEX Associativeity 2. (Ax + My)z = A(XZ) + H()Z), A, HE F, X,y.ZEA Left Linearity 3. x(ly + Uz) = A(xy) + H(XZ), A, He F, XYZCA Right Linearity Definition 2.42 (Generated Subalgebra). Let C be a nonempty subset of an algebra A, then a(C) : B B is subalgebra of A. CCB is the smallest subalgebra of A containing C called the subalgebra generated by C. Theorem 2.51 (Stone-Weierstrass Theorem (Complex Version)). Let (X,p) be a com- pact metric space and A be a subalgebra of the complex algebra C(X, C) such that (1) 16 A (i. e., A contains constants); (2) Vx.yeX, xty if e A: f(x) #f() (i.e., A separates points); (3) VfeA: feA (i.e., Aself-conjugate). Then A is dense in (C(X, Chp..)