Completeness is a central concept in the study of normed spaces, which are the foundational setting for many learning algorithms. A normed space is said to be complete if every Cauchy sequence in the space converges to a limit that is also within the space. Recall that a Cauchy sequence is a sequence of elements in which the distance between successive elements becomes arbitrarily small as the sequence progresses. Formally, a sequence $(x_n)$ in a normed space $(X, \|\cdot\|)$ is Cauchy if for every $\epsilon > 0$, there exists $N$ such that for all $m, n > N$, $\|x_n - x_m\| < \epsilon$. If every Cauchy sequence in $X$ converges to an element of $X$, then $X$ is called a Banach space. Thus, Banach spaces are complete normed spaces, and completeness ensures that limit processes, which are fundamental in analysis and machine learning, always yield elements within the original space.

A fundamental result in functional analysis is that every finite-dimensional normed space is a Banach space. That is, in finite dimensions, all Cauchy sequences converge to a point in the space, regardless of the norm chosen. The proof relies on the fact that all norms on a finite-dimensional vector space are equivalent, meaning that convergence and Cauchy properties do not depend on the specific norm. In practical terms, this means that spaces like $\mathbb{R}^n$ with any norm (such as the Euclidean norm or the maximum norm) are always complete. However, in infinite-dimensional spaces, such as spaces of continuous functions or sequence spaces, completeness is not automatic. Some infinite-dimensional normed spaces are not complete, and their completion (the process of adding limits of all Cauchy sequences) is necessary to obtain a Banach space. This distinction is crucial in advanced machine learning, where hypothesis spaces may be infinite-dimensional, and completeness must be checked or enforced to guarantee the soundness of learning algorithms.