In functional analysis a partial isometry is a linear map between Hilbert spaces such that it is an isometry on the orthogonal complement of its kernel. The orthogonal complement of its kernel is called the initial subspace and its range is called the final subspace. Partial isometries appear in the polar decomposition. General The concept of partial isometry can be defined in other equivalent ways. If U is an isometric map defined on a closed subset H1 of a Hilbert space H then we can define an extension W of U to all of H by the condition that W be zero on the orthogonal complement of H1. Thus a partial isometry is also sometimes defined as a closed partially defined isometric map. Partial isometries (and projections) can be defined in the more abstract setting of a semigroup with involution; the definition coincides with the one herein. In finite-dimensional vector spaces, a matrix is a partial isometry if and only if is the projection onto its support. Equivalently, any finite-dimensional partial isometry can be represented, in some choice of basis, as a matrix of the form , that is, as a matrix whose first columns form an isometry, while all the other columns are identically 0. Yet another general way to characterize finite-dimensional partial isometries is to observe that partial isometries coincide with the Hermitian conjugates of isometries, meaning that a given is a partial isometry if and only if is an isometry. More precisely, if is a partial isometry, then is an isometry with support the range of , and if is some isometry, then is a partial isometry with support the range of . Operator Algebras For operator algebras one introduces the initial and final subspaces: C*-Algebras For C*-algebras one has the chain of equivalences due to the C*-property: So one defines partial isometries by either of the above and declares the initial resp. final projection to be W*W resp. WW*. A pair of projections are partitioned by the equivalence relation: It plays an important role in K-theory for C*-algebras and in the Murray-von Neumann theory of projections in a von Neumann algebra. Special Classes Projections Any orthogonal projection is one with common initial and final subspace: Embeddings Any isometric embedding is one with full initial subspace: Unitaries Any unitary operator is one with full initial and final subspace: (Apart from these there are far more partial isometries.)