ABSTRACT
Analysis is the source of several powerful techniques in applied mathematics. This chapter discusses basic concepts in analysis, and complex analysis and outlines asymptotic behavior of algorithms. It introduces concepts such as fields, vector spaces over fields, linear mappings, and tensor products and describes dot product, vector product, and normed and complete vector spaces. The chapter presents concepts such as completeness, compactness, and orthogonality and traces Hilbert spaces, nonorthogonal expansion of functions, and biorthogonal bases. It examines notions such as neighborhoods, interior points, interior of a set, exterior point, boundary points, limit points, open set, closure of a set, closed set, dense set, and compact set. The definitions of neighborhoods, limit points, closed sets, bounded sets, interior and exterior points, boundary points, and open sets in the complex plane are similar to those defined on the real line. Study of algorithmic-complexity classes helps in classifying the algorithms based upon their complexity.
