Permutations, combinations, binomial theorem, magic squares, and partition theory can combine into one subject {combinatorial analysis}.
Continuous functions, convergences, and limits can combine into one subject {functional analysis}. Analysis includes infinite series, ordinary differential equations, partial differential equations, differential geometry, calculus, and calculus of variations. Analysis excludes plane geometry, solid geometry, and computational methods. Analysis uses arithmetic, variable, function, continuity, differentiability, integrability, limits, infinitesimals, infinite, least upper bound, uniformity, convergence, and fundamental theorem of calculus.
purposes
Functional analysis can be for generalized moment problem, statistical mechanics, fixed-point theorems, partial-differential-equation existence and uniqueness theorems, calculus of variations, and continuous compact-group representation. Linear functional analysis can study integral equations.
functional
Functions of functions {functional} map function to number. Functionals can find areas of products of two functions, over intervals. Functionals can evaluate functions at points. Functionals have generalized derivatives that are also functionals, but can have singularities.
Sum from p = 1 to p = infinity of (z(p) * Z(p))^0.5, where Z(p) is z(p) complex conjugate, can study calculus of variations {linear functional analysis}.
Analysis {standard analysis} can use limits and exhaustion method. Standard analysis and nonstandard analysis use same language and rules, but interpretation is different.
Analysis {nonstandard analysis} can use infinitesimals that can never get large. In nonstandard analysis, numbers, metrics, and spaces always have nearby values.
contradiction
Non-standard analysis can introduce contradictions, because added infinitesimals do not necessarily stay small. Infinitesimals can violate Archimedes principle {non-Archimedian}.
theorem
For proposition sets, if finite proposition subsets are true in standard analysis, whole proposition set is true in nonstandard analysis {compactness theorem}.
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Date Modified: 2022.0225