Are polynomials dense in C01 Solutionyes polynomials are den

Solution

yes polynomials are dense in C(0,1).

As a consequence of the Weierstrass approximation theorem, one can show that the space C[a, b] is separable: the polynomial functions are dense, and each polynomial function can be uniformly approximated by one with rational coefficients; there are only countably many polynomials with rational coefficients. Since C[a, b] is Hausdorff and separable it follows that C[a, b] has cardinality equal to 2^₀ — the same cardinality as the cardinality of the reals. (Remark: This cardinality result also follows from the fact that a continuous function on the reals is uniquely determined by its restriction to the rationals.