W a continuous linear map. If dimc(Wju(V)) < 00, then u(V) is closed in W.

Proposition. Let 'P be a form of type (0, 1) on the compact Riemann surface X. Then, 3f E COO(X) such that 8f = 'P if and only if, for every holomorphic 1-form w on X, we have 10. The Riemann-Roch Theorem and some Applications Throughout this section, X will be a compact Riemann surface. We begin with some definitions. Let D = 2:~=1 niPi be a divisor on X. The integer d D [and written degD]. = 2:~=1 ni is called the degree of Let W be a meromorphic I-form, w "¥=- O. If a E X, we denote by resa(w) (the residue at a of w) the following: If (U, z) is a local coordinate with z(a) = 0, and w = f dz, then resa(w) = residue of f at a = the coefficient of ~ in the Laurent expansion 2:~N cvz v of f.