The widespread applications and the foundations of the success of deep learning methods play a fundamental role in understanding deep learning and expanding application fields. Starting from the basic elements of sample space, model space, empirical risk minimization, and regularization in machine learning, this talk explores their special forms in the field of deep learning, covering the density of deep neural network functions in continuous function space, the regularized subspace of neural networks achieved through invariance, stability, and local invariance, and the development framework of the existence and approximation of functions driven by the mathematical structures of applications.