| Abstract [eng] |
In this thesis we delve into the fundamental concepts of bounded linear operators and isomorphisms in the context of metric and normed spaces, providing insights into their properties, applications, and significance in functional analysis. Bounded linear operators serve as essential tools for studying the relationship between different metric and normed spaces. These operators are linear mappings that preserve the algebraic structure of vector spaces. This thesis explores the definition and properties of bounded linear operators, emphasizing their role in preserving idstances and geometric properties between normed spaces. Isomorphisms, on the other hand, establish a one-to-one correspondence between elements of different normed spaces while preserving their algebraic structure. This thesis investigates the properties of isomorphisms and their significance in functional analysis, particularly in establishing equivalence between metric and normed spaces. Through detailed examples and theoretical discussions, we illustrate how bounded linear operators and isomorphisms contribute to the development of functional analysis and provide powerful tools for studying the properties of metric and normed spaces. It explores applications of these concepts in various mathematical contexts, ranging from topology and geometry to optimization and mathematical physics. |
