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Fredholm's Theorems
In mathematics, Fredholm's theorems are a set of celebrated results of Ivar Fredholm in the Fredholm theory of integral equations. There are several closely related theorems, which may be stated in terms of integral equations, in terms of linear algebra, or in terms of the Fredholm operator on Banach spaces. The Fredholm alternative is one of the Fredholm theorems. Linear algebra Fredholm's theorem in linear algebra is as follows: if ''M'' is a Matrix (mathematics), matrix, then the orthogonal complement of the row space of ''M'' is the null space of ''M'': :(\operatorname M)^\bot = \ker M. Similarly, the orthogonal complement of the column space of ''M'' is the null space of the adjoint: :(\operatorname M)^\bot = \ker M^*. Integral equations Fredholm's theorem for integral equations is expressed as follows. Let K(x,y) be an integral kernel, and consider the Homogeneous polynomial, homogeneous equations :\int_a^b K(x,y) \phi(y) \,dy = \lambda \phi(x) and its complex adjoint : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
