|
σ-unital Algebra
In mathematics, particularly in functional analysis and ring theory, an approximate identity is a net (mathematics), net in a Banach algebra or ring (mathematics), ring (generally without an identity) that acts as a substitute for an identity element. Definition A right approximate identity in a Banach algebra ''A'' is a net \ such that for every element ''a'' of ''A'', \lim_\lVert ae_\lambda - a \rVert = 0. Similarly, a left approximate identity in a Banach algebra ''A'' is a net \ such that for every element ''a'' of ''A'', \lim_\lVert e_\lambda a - a \rVert = 0. An approximate identity is a net which is both a right approximate identity and a left approximate identity. C*-algebras For C*-algebras, a right (or left) approximate identity consisting of self-adjoint elements is the same as an approximate identity. The net of all positive elements in ''A'' of norm ≤ 1 with its natural order is an approximate identity for any C*-algebra. This is called the canonical approxi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
