A study on the teaching reform model of AI-assisted Real Analysis and Functional Analysis: Taking the dominated convergence theorem and the Hahn–Banach theorem as examples
人工智能辅助《实变函数与泛函分析》课程教学改革的模式探讨——以控制收敛定理与 Hahn–Banach 定理为例

摘要

随着生成式人工智能（AI）、智慧教学平台和课堂互动工具的发展，高阶数学理论课程教学改革获得了新的技术支持。《实变函数与泛函分析》是典型的证明型高阶数学课程，教学中存在概念理解难、条件辨析难、证明结构复杂和个性化支持不足等问题。针对上述教学痛点，本文以“教师主导、学生主体、平台支撑、AI 辅助”为基本理念，构建“智慧树—雨课堂—AI大模型”协同支持的“课前—课中—课后”一体化教学模式。本文选取控制收敛定理和 Hahn–Banach 定理作为典型案例，展示该模式在条件辨析和抽象证明教学中的应用，以期为《实变函数与泛函分析》及同类证明型数学课程的智能化教学改革提供参考。

Abstract

With the development of generative artificial intelligence (AI), smart teaching platforms, and classroom interaction tools, new technical support has become available for the teaching reform of advanced theoretical mathematics courses. Real Analysis and Functional Analysis is a typical proof-based advanced mathematics course. In its teaching process, students often face difficulties such as understanding abstract concepts, distinguishing theorem conditions, grasping complex proof structures, and receiving insufficient personalized learning support. In response to these teaching challenges, this paper follows the basic principle of “teacher guidance, student-centered learning, platform support, and AI assistance,” and constructs an integrated “pre-class, in-class, and post-class” teaching model supported by the collaboration of Zhihuishu, Rain Classroom, and AI large models. Taking the Dominated Convergence Theorem and the Hahn–Banach Theorem as representative cases, this paper demonstrates the application of the proposed model in condition-discrimination teaching and abstract-proof teaching, aiming to provide a reference for the intelligent teaching reform of Real Analysis and Functional Analysis and similar proof-based mathematics courses.