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For compact X, L(D) is finite dim
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Write D= P - N, nonnegative with disjoint support. We prove by induction on deg P. When deg P = 0, then as P is nonnegative I.e. All coefficient are nonnegative, P= 0. Then L(P) = L(0) is the the space of constant functions so must be of dim 1 ( because if div(f) geq 0 then all doff nonnegative but deg div f = 0, so all coefff is 0 and so f has no poles or zeroes. Thus f is holomorphic on a compact Riemann surface so f must be constant). As L(D) is a subspace of L(P), we must have dim L(D) is leq 1 which is leq 1 + deg (P) Suppose we prove that dim L(D) leq 1 + deg P for deg P = k-1. Now suppose deg P = k. Apply induction hypothesis for D-q and P- q where D(q) is at least 1. Then remember that dim L(D) leq dim L(D-q) + 1
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