Complex Bounded Operators in Isabelle/HOL
This work provides foundational formalization for complex vector spaces and bounded operators in Isabelle/HOL, benefiting researchers and practitioners using interactive theorem proving for quantum mechanics and functional analysis.
The paper presents a formalization of bounded operators on complex vector spaces in Isabelle/HOL, including definitions and theorems for unitaries, projectors, adjoints, and more, with code generation support for finite-dimensional operators.
We present a formalization of bounded operators on complex vector spaces in Isabelle/HOL. Our formalization contains material on complex vector spaces (normed spaces, Banach spaces, Hilbert spaces) that complements and goes beyond the developments of real vectors spaces in the Isabelle/HOL standard library. We define the type of bounded operators between complex vector spaces (cblinfun) and develop the theory of unitaries, projectors, extension of bounded linear functions (BLT theorem), adjoints, Loewner order, closed subspaces and more. For the finite-dimensional case, we provide code generation support by identifying finite-dimensional operators with matrices as formalized in the Jordan_Normal_Form AFP entry.