Locally Convex Spaces

Banach Space Theory, Mathematical Physics, and Distribution Theory Applications

Thomas E. Gilsdorf

• Publisher: De Gruyter
• Year: 2026
• Print ISBN:
  9783111390796
• Edition: 1
• eBook ISBN:
  9783111394152
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

This book provides an updated introductory textbook on the topic of locally convex topological vector spaces. Readers who enjoy combinations of topology, analysis and linear algebra will discover fascinating relevance in this work. The target audience is anyone, such as graduate students and researchers who have a typical background in one semester courses in topics like topology, and analysis of normed/Banach spaces. Several chapters are standard, with topics like basic properties of topological vector spaces and local convexity, the Hahn-Banach Theorem, the basics of duality, and some deep results such as the Banach-Steinhaus and Closed Graph theorems. Unique features of this book include the inclusion of examples from normed and Banach spaces that show how such spaces fit in within the larger context of locally convex spaces, the utility of sequences and series, examples of recent applications to results such as in distribution theory and optimization, and introductions to some atypical contexts of locally convex spaces such as convergence vector spaces and abstract duality pairs. Readers will find a colorful world of spaces and results that will be useful for starting research, or gaining insight into analysis and topology.

Table of contents

• Title Page
• Copyright
• Contents
• Frontmatter
• Dedication
• Contents
• 1 Topological vector spaces: Basic ideas
• 2 Locally convex spaces: The beginning
• 3 Creating locally convex spaces
• 4 Duality
• 5 Some special types of locally convex spaces
• 6 Sequences and series in locally convex spaces
• 7 Local completeness
• 8 The closed graph theorem
• 9 Applying concepts of locally convex spaces
• 10 Pathways to research
• 11 Other resources
• Index
• Subject Index