This chapter introduces the fundamental concepts of topological vector spaces (T.V.S.). It emphasizes the generality of the scalar field \( K \), which can be any non-discrete valuated field, to highlight essential properties. The chapter is divided into several sections:
1. **Vector Space Topologies**: Discusses the definition of a T.V.S. and the axioms that ensure the continuity of addition and scalar multiplication.
2. **Construction of New T.V.S.**: Provides methods for creating new T.V.S. from existing ones.
3. **Finite-Dimensional Spaces**: Compiles standard tools for working with finite-dimensional spaces.
4. **Affine Subspaces and Hyperplanes**: Explains the concepts of affine subspaces and hyperplanes.
5. **Boundedness**: Introduces the important concept of boundedness in T.V.S.
6. **Metrizability**: Examines the property of metrizability and its significance in various contexts.
7. **Real to Complex Fields**: Discusses the transition between real and complex fields in T.V.S.
The chapter also includes a detailed proof of several key properties of T.V.S., such as the homeomorphism of translations, the closure of subsets, and the openness of sums of open subsets. These properties are crucial for understanding the structure and behavior of T.V.S.This chapter introduces the fundamental concepts of topological vector spaces (T.V.S.). It emphasizes the generality of the scalar field \( K \), which can be any non-discrete valuated field, to highlight essential properties. The chapter is divided into several sections:
1. **Vector Space Topologies**: Discusses the definition of a T.V.S. and the axioms that ensure the continuity of addition and scalar multiplication.
2. **Construction of New T.V.S.**: Provides methods for creating new T.V.S. from existing ones.
3. **Finite-Dimensional Spaces**: Compiles standard tools for working with finite-dimensional spaces.
4. **Affine Subspaces and Hyperplanes**: Explains the concepts of affine subspaces and hyperplanes.
5. **Boundedness**: Introduces the important concept of boundedness in T.V.S.
6. **Metrizability**: Examines the property of metrizability and its significance in various contexts.
7. **Real to Complex Fields**: Discusses the transition between real and complex fields in T.V.S.
The chapter also includes a detailed proof of several key properties of T.V.S., such as the homeomorphism of translations, the closure of subsets, and the openness of sums of open subsets. These properties are crucial for understanding the structure and behavior of T.V.S.