The theory of convex sets in Euclidean spaces can be in a natural way transferred to the noncommutative setting of matrix spaces. In this master's thesis we discuss matrix convex sets, their properties and examples, we deal with matrix extreme points and introduce matrix exposed points. We also aspire to understand how much the results in the matrix world resemble those from the classical theory, such as the Krein-Milman theorem. As a device to prove the matricial analogue of the Krein-Milman theorem we explain the Hahn-Banach theorem for matrix convex sets.
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