- Lecturer: Luc Nguyen
Part A Integration is essential, particularly convergence theorems, the theorems of Fubini and Tonelli, the notions of measurable functions, integrable functions, null sets, Lᵖ spaces. Familiarity with the construction of Lebesgue's integral is desirable. No knowledge is needed of outer measure. A good working knowledge of Part A Core Analysis (both metric spaces and complex analysis) is expected. Part B B4.1 Functional Analysis I is desirable, in particular notions of normed vector spaces, Banach spaces and bounded linear operators between them. Taking this course alongside Part B B4.2 Functional Analysis II is recommended, but no knowledge of that course is expected.
The course provides an introduction to real analysis and is a natural continuation of Part A Integration. The techniques and examples studied in this course support, in essential ways, many advanced courses, in particular in analysis and in the study of partial differential equations, and also have applications in mathematical physics and other areas.
By the end of the course, students will be able to
have a deeper understanding of the spaces Lᵖ and their properties,
use the definition of Hardy-Littlewood's maximal function and prove basic results about it,
state, prove, and apply Lebesgue's differentiation theorem,
be familiarised with the notion of convolution, approximations to the identity, and prove results about them,
define and check Gateaux and Fréchet differentiability of functions defined on normed vector spaces, notably Lᵖ spaces,
state and apply the Inverse function theorem and the Implicit function theorem (proof in the finite dimensional case).
Revision of the construction of Lebesgue's integral.
Lebesgue's differentiation theorem.
Lebesgue Lᵖ spaces: completeness, density, uniform convexity, linear functionals and weak convergence.
Convolution, approximation to the identity.
Hardy-Littlewood's maximal function and maximal inequality.
Continuity, Gateaux and Fréchet differentiability of functions on normed vector spaces.
The Implicit function theorem (proof in the finite dimensional case).