# 18.1x

## Analysis

This is the study of functions, as begun in calculus. At MIT this specialty is entered through a single gateway subject: 18.100. For many students 18.100 is the first course requiring them to construct and write out carefully reasoned proofs.

**18.100 (Real Analysis)**Because of the diversity of backgrounds and needs of our students, 18.100 comes in several variants. The variants are equivalent as prerequisites. In addition to the analysis courses listed below, 18.100 is a prerequisite for 18.504, 18.701, and 18.901.- Option A chooses less abstract definitions and proofs, and gives applications where possible.
- Option B is more demanding and is for students with more mathematical maturity; it places more emphasis on point-set topology and n-space, whereas Option A is concerned primarily with the real line.
- Option P is a 15-unit (4-0-11) variant of Option A, with further instruction and practice in written and oral communication. It carries CI-M credit.
- Option Q (formerly 18.100C) is a 15-unit (4-0-11) variant of Option B, with further instruction and practice in written and oral communication. It carries CI-M credit.

**18.101 (Analysis and Manifolds)**This subject frees calculus from the strictures of Euclidean space. It is followed by 18.952 (Theory of Differential Forms), which completes the theoretical treatment of Stokes' theorem. Prerequisites: 18.100 and Linear Algebra.**18.102 (Functional Analysis)**This course studies spaces of functions and culminates with the spectral theorem. Prerequisites: 18.100 and Linear Algebra.**18.103 (Fourier Analysis)**Lebesgue integration with applications to probability and to Fourier series and integrals. Prerequisites: 18.100 and Linear Algebra.**18.104 (Seminar in Analysis)**Students present and discuss subject matter taken from current journals or books. Topics vary from year to year. Instruction and practice in written and oral communication provided. Prerequisite: 18.100. This class carries CI-M credit.**18.112 (Complex Analysis)**This course provides a rigorous treatment of complex variables, more theoretical than 18.04. Prerequisites: 18.100 and Linear Algebra.**18.152 (Partial Differential Equations)**A rigorous treatment of linear PDEs, with an introduction to some nonlinear examples. The differential equations treated in this course are essentially the same as those studied in 18.303, but the power of 18.100 allows a deeper treatment. Prerequisites: 18.100 and Linear Algebra.