The spectrum of a linear operator that operates on a Banach space is a fundamental concept of functional analysis. The spectrum consists of all scalars such that the operator does not have a bounded inverse on . The spectrum has a standard decomposition into three parts: a point spectrum, consisting of the eigenvalues of ; a continuous spectrum, consisting of the scalars that are not eigenvalues but make the range of a proper dense subset of the space; a residual spectrum, consisting of all other scalars in the spectrum. This decomposition is relevant to the study of differential equations, and has applications to many branches of science and engineering. A well-known example from quantum mechanics is the explanation for the discrete spectral lines and the continuous band in the light emitted by excited atoms of hydrogen. Decomposition into point spectrum, continuous spectrum, and residual spectrum For bounded Banach space operators Let X be a Banach space, B(X) the family of bounded operators on X, and . By definition, a complex number λ is in the spectrum of T, denoted σ(T), if does not have an inverse in B(X). If is one-to-one and onto, i.e. bijective, then its inverse is bounded; this follows directly from the open mapping theorem of functional analysis. So, λ is in the spectrum of T if and only if is not one-to-one or not onto. One distinguishes three separate cases: is not injective. That is, there exist two distinct elements x,y in X such that . Then is a non-zero vector such that . In other words, λ is an eigenvalue of T in the sense of linear algebra. In this case, λ is said to be in the point spectrum of T, denoted . is injective, and its range is a dense subset R of X; but is not the whole of X. In other words, there exists some element x in X such that can be as close to x as desired, with y in X; but is never equal to x. It can be proved that, in this case, is not bounded below (i.e. it sends far apart elements of X too close together). Equivalently, the inverse linear operator , which is defined on the dense subset R, is not a bounded operator, and therefore cannot be extended to the whole of X. Then λ is said to be in the continuous spectrum,