To understand generalization in machine learning from a functional-analytic perspective, you can frame the problem in terms of operators between function spaces. Consider a hypothesis space as a subset of a Banach or Hilbert space, where each hypothesis corresponds to a function mapping inputs to outputs. The learning process can be viewed as applying an operator that maps data (for example, empirical distributions or sample points) to hypotheses. Generalization is then concerned with how well this operator transfers information from finite samples to the underlying data-generating distribution. Formally, generalization requires that the operator is not only continuous but, ideally, compact: a compact operator ensures that bounded sequences of hypotheses (for example, those selected by minimizing empirical risk) have convergent subsequences in the hypothesis space. This property is crucial because it links the stability of learning (small changes in data lead to small changes in the learned function) with the ability to extract meaningful patterns that persist beyond the training data.

Regularization methods, such as adding penalty terms to loss functions or restricting the complexity of hypotheses, can be interpreted as strategies to enforce compactness or continuity in the hypothesis space. By constraining the set of admissible hypotheses (for instance, through norm penalties or explicit bounds), regularization ensures that the learning operator does not select wildly oscillating or overly complex functions that fit the training data but fail to generalize. In functional analysis, such constraints often correspond to precompactness or compactness conditions, which guarantee that every sequence of hypotheses has a convergent subsequence with respect to the topology of the space. This perspective clarifies why regularization improves generalization: it shapes the hypothesis space so that the learning process is governed by well-behaved, stable mappings, making overfitting less likely.