During the past decades much of finite-dimensional systems theory has been generalized to infinite dimensions. However, there is one important flaw in this theory: it guarantees only complex solutions, even when the data is real. We show that the standard solutions of many classical problems with real data are also real. We call a (possibly matrix-or operator-valued) holomorphic function G real (real-symmetric) if G((z) over bar) = <(G(z))over bar> for every z. We show that if such a function can be presented as G = NM-1, where N, M is an element of H-infinity, then we have G = NRMR-1 , where N-R, M-R is an element of H-infinity are real and weakly right coprime. Consequently, if a real function G has a stabilizing compensator (i.e., a function K such that [(I)(-G) (K)(I)](-1) is an element of H-infinity), then G has a real doubly coprime factorization and a Youla parameterization of all real stabilizing controllers. If a system of the form <(x)over dot> = Ax + Bu, y = Cx + Du or of the form x(n+1) = Ax(n) + Bu-n, y(n) = Cx(n) + Du(n) has real (possibly unbounded, constant) coefficients A, B, C, and D, then the system is stabilizable iff it is stabilizable by a real state-feedback operator. This holds for both exponential stabilization and output stabilization. A real stabilizing state-feedback operator is then given by the standard linear quadratic regulator (LQR) feedback operator, hence the standard (complex) formulae can be used to find this real solution. Analogous results are established for other optimization, factorization, approximation, and representation problems too.