The development of real variable function is late, and integral theory is an important part of it. As a generalization of the concept of line segment length, tolerance and measure are introduced, thus the concept of integral is generalized.
Because mathematics is systematic, continuous, abstract, rigorous and enlightening, mathematics learning should be meaningful discovery learning.
Functional analysis is a branch of studying the mapping from topological linear space to topological linear space satisfying various topological and algebraic conditions. It is developed from the study of variational problems, integral equations and theoretical physics. It comprehensively uses the viewpoints of function theory, geometry and modern mathematics to study functions, operators and limit theory in infinite dimensional vector space. It can be regarded as analytic geometry and mathematical analysis of infinite dimensional vector space.
Mathematics learning, in addition to ensuring a solid mathematical foundation, should also think like a mathematician as much as possible. It is necessary to link mathematical knowledge through mathematical thinking, ideas and methods to build a stable knowledge system.
With the development of calculus and the description of the movement law of the objective material world, the development of ordinary differential equations and partial differential equations has been promoted. With the development of the research fields of physical science and engineering technology, the application scope of differential equations is more and more extensive. Conversely, from the point of view of mathematics itself, the solution of partial differential equations promotes the development of functional theory, variational method, series expansion, ordinary differential equations, algebra, differential geometry and other disciplines. From this perspective, partial differential equations have become the center of mathematics.