École polytechnique learning platform

Prerequisite: MAA202

MAA301 is devoted to the modern theory of integration. After first constructing the Lebesgue integral, and explaining how it improves the Riemann integral, a major part of the course will be devoted to discovering the power and ease of use of this tool.

Applications in probability theory will then be briefly described. The course will finally provide an introduction to Lebesgue spaces and the Fourier transform, in order to demonstrate the usefulness of the theory for applications in physics and economics.

After constructing the Lebesgue integral, and explaining how it includes the Riemann integral, a long part of the course will be devoted to discover what a powerful and easy to use tool it is. First applications will be given to probability theory. Finally, Lebesgue spaces and the Fourier transform will be introduced to demonstrate the usefulness of the theory in applications such as physics and economy.

Prerequisite: MAA202

MAA302 is devoted first to the theory of metric and topological spaces in an abstract setting, including numerous examples of function spaces. We will then shift our focus towards Banach spaces, motivated by applications in optimization. Following this, the course will examine differentiable functions, smooth functions, and their local properties. Restricting our attention to finite dimensional spaces, the course will conclude with an abstract theory of optimization, with applications in economics and physics: optimization without constraints and with constraints, and the well-known Lagrange multiplier theorem will all be studied in detail.