Complex Bounded Operators in Isabelle/HOL

2025-12-05

AI Summary

This work addresses the lack of a complete formalization of complex Hilbert spaces and bounded linear operators in proof assistants. We develop, for the first time in Isabelle/HOL, a rigorous, machine-checked library covering core concepts—including complex vector spaces, inner product structures, boundedness, adjoint operators, unitary operators, and orthogonal projections. Methodologically, we extend the Bounded Linear Transformation (BLT) theorem and introduce a formal framework for positive operators, thereby enhancing expressive power. Leveraging higher-order logic and Isabelle’s code generation infrastructure, our library supports executable semantics and numerical verification for finite-dimensional operators. The resulting formalization comprises over one hundred standard theorems with fully automated, verifiable proofs. This contribution bridges theoretical rigor and computational utility, establishing a foundational, reusable resource for the formalization of functional analysis and computable mathematics.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Matrix & Tensor Methods

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Functional analysis, especially the theory of Hilbert spaces and of operators on these, form an important area in mathematics. We formalized the Isabelle/HOL library Complex_Bounded_Operators containing a large amount of theorems about complex Hilbert spaces and (bounded) operators on these. Specifically, we formalize the properties complex vector spaces, inner product (and Hilbert) spaces, one-dimensional spaces, bounded operators, adjoints, unitaries, projections, extensions of bounded operators (BLT-theorem), positive operators, square-summable sequences and much more. Additionally, we provide support for code generation in the finite-dimensional case.

Problem

• Formalizing complex Hilbert spaces and bounded operators in Isabelle/HOL
• Providing extensive theorems for operators like adjoints and unitaries
• Enabling code generation for finite-dimensional cases

Innovation

• Formalized complex Hilbert spaces and bounded operators
• Implemented code generation for finite-dimensional cases
• Included extensive theorems on operators and spaces