Eynstone said:Yes, it's due to the compactness of the embedding.
Compact embedding is a concept in functional analysis that describes the relationship between two topological spaces, where one space is embedded into the other in a compact manner. In other words, a compact embedding is a way of mapping a larger space onto a smaller one while preserving certain properties.
Compact embedding and strong convergence are two distinct concepts in functional analysis. Compact embedding refers to the relationship between two topological spaces, while strong convergence refers to the convergence of a sequence of elements in a space to a specific limit point. However, compact embedding can be used to prove strong convergence in certain cases.
Compact embedding has various applications in mathematics, particularly in functional analysis and differential equations. It is used to prove the existence and uniqueness of solutions to certain problems, as well as to establish important properties of these solutions, such as regularity and convergence.
One example of compact embedding is the Sobolev embedding theorem, which states that certain functions in a Sobolev space can be continuously embedded into a Lebesgue space. Another example is the Arzelà–Ascoli theorem, which describes the compactness of equicontinuous and uniformly bounded sequences of functions.
Studying compact embedding and strong convergence allows for a deeper understanding of the relationship between different topological spaces and their properties. It also has practical applications in various fields of mathematics, such as in the analysis of partial differential equations and in the study of dynamical systems.