Selfadjoint endomorphisms of inner product spaces are defined and studied. Any selfadjoint endomorphism of a finitely-generated inner product space is shown to have a nonempty spectrum. The notion of orthogonal decomposition of an endomorphism is introduced. Selfadjoint endomorphisms of finitely-generated inner product spaces are shown to be orthogonally diagonalizable, with the converse true for spaces over the real numbers. Positive-definite endomorphisms are introduced and characterized. Application is made to Cholesky decompositions. Isometries of finitely-generated inner product spaces are studied and characterized.