Banach Space

Banach spaces, infinite-dimensional vector spaces with a specific distance metric, are central to advanced mathematical analysis and increasingly important in machine learning and scientific computing. Current research focuses on extending deep learning techniques, such as transformers and neural integral operators, to approximate and learn operators acting between these spaces, often leveraging methods like Leray-Schauder mappings and orthogonal projections. This work addresses challenges in decentralized control, optimal learning, and solving inverse problems, with implications for diverse fields including PDE modeling, reinforcement learning, and signal processing.