Hahn–Banach theoremThe Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting". Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem, and has numerous uses in convex geometry.
Measure (mathematics)In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge.
Weak convergence (Hilbert space)In mathematics, weak convergence in a Hilbert space is convergence of a sequence of points in the weak topology. A sequence of points in a Hilbert space H is said to converge weakly to a point x in H if for all y in H. Here, is understood to be the inner product on the Hilbert space. The notation is sometimes used to denote this kind of convergence. If a sequence converges strongly (that is, if it converges in norm), then it converges weakly as well.
Banach limitIn mathematical analysis, a Banach limit is a continuous linear functional defined on the Banach space of all bounded complex-valued sequences such that for all sequences , in , and complex numbers : (linearity); if for all , then (positivity); where is the shift operator defined by (shift-invariance); if is a convergent sequence, then . Hence, is an extension of the continuous functional where is the complex vector space of all sequences which converge to a (usual) limit in .
Sturm–Liouville theoryIn mathematics and its applications, classical Sturm–Liouville theory (developed by Joseph Liouville and Jacques Charles François Sturm) is the theory of real second-order linear ordinary differential equations of the form: for given coefficient functions , , and , an unknown function of the free variable , and an unknown constant . All homogeneous (i.e. with the right-hand side equal to zero) second-order linear ordinary differential equations can be reduced to this form.