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) The systematic study of real- and complex-valued continuous functions. (2) In logic, second-order arithmetic.
Bernoulli Number	One of a sequence of signed rational numbers that can be defined using a certain simple generating function. They are very important in number theory and analysis, and commonly arise in series expansions of trigonometric functions.
Calculus of Variations	A generalization of 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	An example of an uncountable set of measure zero.
Convolution	An integral that expresses the amount of overlap of one function g as it is shifted over another function f.
Delta Function	A generalized function that has the property that the convolution integral of it with any function equals the value of that function at zero.
Fourier Series	An expansion of a periodic function in terms of an infinite sum of sines and cosines.
Gamma Function	An extension of the factorial to real and complex arguments.
Lebesgue Measure	An extension of the classical notions of length and area to more complicated sets.
Measure	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 vector space that has a complete norm. Banach spaces are important in the study of infinite-dimensional vector spaces.
Functional Analysis:	A branch of mathematics concerned with infinite-dimensional vector spaces and mappings between them.
Hilbert Space:	A vector space that has a complete inner product. Hilbert spaces are important in the study of infinite-dimensional vector spaces. MathWorld