1. Firstly, Lp space is a space composed of integrable functions of degree p, where p is a real number greater than 1. In Lp space, the integration of functions is endowed with important significance. The definition and properties of Lp space were put forward by French mathematician André Weil in AD 1930, and later extended and deepened by Polish mathematician Joseph Masinkiewicz.
2. Thirdly, L∞ space is a space composed of all bounded functions (that is, the range of function values is in a finite interval). In L∞ space, the infinity of functions is endowed with great significance. The definition and properties of L∞ space were put forward by American mathematician paul halmos in AD1940s, and later extended and deepened by French mathematician J.-P. Kahane.
3. Secondly, the relationship between Lp space and L∞ space is very close. Firstly, Lp space can be regarded as a subset of L∞ space. This is because if a function is a function in Lp space, it is also a function in L∞ space. Therefore, Lp space can be regarded as a subset of L∞ space.
4. Secondly, the relationship between Lp space and L∞ space can also be reflected by their properties. For example, the function in Lp space is square integrable, while the function in L∞ space is bounded. Therefore, the properties of Lp space and L∞ space are different, but they are interrelated.
5. Finally, the relationship between Lp space and L∞ space can be reflected by their applications. For example, Lp space is widely used in probability theory, statistics, physics, engineering and other fields, while L∞ space is widely used in signal processing, image processing, control theory and other fields. Therefore, Lp space and L∞ space have different applications in different fields.
Introduction to functional analysis
Functional analysis is a branch of mathematics, which studies the function space and the properties of functions. It involves many fields such as function theory, geometry, topology, algebra, mathematical analysis and so on. Functional analysis is an important branch of modern mathematics, and its research objects are mainly functions and spaces.
There are two main research objects of functional analysis: one is function space, and the other is functional.
Function space is a space composed of all functions satisfying certain properties, such as Lp space, L∞ space, Hilbert space and so on. Functional is a mathematical object that operates on functions, such as integral, derivative, dot product, etc.
The basic principles and methods of functional analysis mainly include: topological properties of space, continuity and differentiability of function, structure and properties of linear space, properties and calculation of functional, measure theory and integral theory.