Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, particularly functional analysis a projection-valued measure (PVM) is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a Hilbert space. Projection-valued measures are used to express results in spectral theory, such as the spectral theorem for self-adjoint operators. The idea of a projection-valued measure is generalized by the positive operator-valued measure, where the need for the orthogonality implied by projection operators is replaced by the idea of a set of operators that are a non-orthogonal partition of unity. This generalization is motivated by applications to quantum information theory. In mathematics, functional analysis is the branch of analysis concerned with vector spaces in which limit processes can be defined, and the linear operators acting upon these spaces that are (in some way) compatible with these limits.

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