Prof in vector space of continuity of Fouriercoefficients of complex series

Help need.

Consider the vector space of limited complex series of the uniform norm.

Prove that the Fouriercoefficient c(f) given by the inner product space (from -phi to phi) of the complex conjugate of two functions (f(x)*g(x), where f(x) just is a function and that g(x) in the conjugate), defines a norm reducing function and conclude from this that this function is continuers.

Hint : use either the Bessel or the Cauchy-Schwarz inequality.