 TOPICS   # Topics in an Analysis Course

To learn more about a topic listed below, click the topic name to go to the corresponding MathWorld classroom page.

### General

 Analysis (1) In higher mathematics, analysis is the systematic study of real- and complex-valued continuous functions. (2) In the mathematical theory of logic, analysis refers to second-order arithmetic. Bernoulli Number A Bernoulli number is one in a sequence of signed rational numbers that can be defined using a certain simple generating function. Bernoulli numbers are very important in number theory and analysis, and commonly arise in series expansions of trigonometric functions. Calculus of Variations The calculus of variations is a generalization of the usual calculus that seeks to find the path, curve, surface, etc., for which a given function has a stationary value (which, in physical problems, is usually a minimum or maximum). Cantor Set A Cantor set is a particular example of an uncountable set of measure zero constructed from a unit interval by recursively removing the middle third of subintervals. Convolution Convolution is the integral transform that expresses the amount of overlap of one function g as it is shifted over another function f. Delta Function The delta function, also called the Dirac delta function, is a generalized function that has the property that its convolution with any function f equals the value of f at zero. Fourier Series A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines. Gamma Function The gamma function is an extension of the factorial to real and complex arguments. Lebesgue Measure Lebesgue measure is an extension of the classical notions of length and area to more complicated sets. Measure A measure is a function that quantifies the size of a subset of a Euclidean space. Measures are used for integration and are important in differential equations and probability theory.

### Functional Analysis

 Banach Space: A Banach space is a vector space that has a complete norm. Banach spaces are important in the study of infinite-dimensional vector spaces. Functional Analysis: Functional analysis is a branch of mathematics concerned with infinite-dimensional vector spaces and mappings between them. Hilbert Space: A Hilbert space is a vector space that has a complete inner product. Hilbert spaces are important in the study of infinite-dimensional vector spaces.