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Forster's result can be seen as the Serre-Swan theorem for Stein manifolds. A proof and many details can be found, among other references, in the thesis of A. S. Morye "On the Serre-Swan Theorem and on Vector Bundles over Real Abelian Varieties". Section $2.3.5$ reviews Forster's result on a few pages. It is more or less deduced from the other results. However, I am not sure, if you find this satisfying. Another discussion in this direction can be found here: Holomorphic vector bundles and Swan's theoremHolomorphic vector bundles and Swan's theorem.

Forster's result can be seen as the Serre-Swan theorem for Stein manifolds. A proof and many details can be found, among other references, in the thesis of A. S. Morye "On the Serre-Swan Theorem and on Vector Bundles over Real Abelian Varieties". Section $2.3.5$ reviews Forster's result on a few pages. It is more or less deduced from the other results. However, I am not sure, if you find this satisfying. Another discussion in this direction can be found here: Holomorphic vector bundles and Swan's theorem.

Forster's result can be seen as the Serre-Swan theorem for Stein manifolds. A proof and many details can be found, among other references, in the thesis of A. S. Morye "On the Serre-Swan Theorem and on Vector Bundles over Real Abelian Varieties". Section $2.3.5$ reviews Forster's result on a few pages. It is more or less deduced from the other results. However, I am not sure, if you find this satisfying. Another discussion in this direction can be found here: Holomorphic vector bundles and Swan's theorem.

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Dietrich Burde
Dietrich Burde
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Forster's result can be seen as the Serre-Swan theorem for Stein manifolds. A proof and many details can be found, among other references, in the thesis of A. S. Morye "On the Serre-Swan Theorem and on Vector Bundles over Real Abelian Varieties". Section $2.3.5$ reviews Forster's result on a few pages. It is more or less deduced from the other results. However, I am not sure, if you find this satisfying. Another discussion in this direction can be found here: Holomorphic vector bundles and Swan's theorem.