The difference between Pure Mathematics and Applied Mathematics lies in application and use. Pure Mathematics are studied with no consideration of necessity or application; they develop the principles of Mathematics for the sake of the principles of Mathematics. Research in the Fibonacci Sequence is an example of this: the Fibonacci Sequence has almost no useful application to mankind. Applied Mathematics, on the other hand, are studied purely for the sake of application. Examples of this lie in Economics, Computer Science, and Engineering.
Occasionally, Pure Maths may lead to a useful application -- research in the field of Physics is usually done without the intention of a useful application; however, what we have learned about physics is used in processes stretching from the creation of personal computers to launching a rocket from Cape Canaveral.
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm,topology, etc.) and the linear operators acting upon these spaces and respecting these structures in a suitable sense.
Real analysis (traditionally, the theory of functions of a real variable) is a branch of mathematical analysis dealing with the real numbers and functions of a real variable.Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigatesfunctions of complex numbers.
Topology (from the Greek τόπος, “place”, and λόγος, “study”) is a major area of mathematics concerned with properties that are preserved undercontinuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing, although the notion of stretching employed in mathematics is not quite the everyday notion: see below and the definition of homeomorphism for details of the mathematical notion. Topology emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation.
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.
Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis. The term "mathematical statistics" is closely related to the term "statistical theory" but also embraces modelling for actuarial science and non-statistical probability theory.
Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series.