Finite Dimensional Real Normed Spaces are Proper Metric Spaces

In this article, we formalize in Mizar [1], [2] the topological properties of finite-dimensional real normed spaces. In the first section, we formalize the Bolzano-Weierstrass theorem, which states that a bounded sequence of points in an n-dimensional Euclidean space has a certain subsequence that converges to a point. As a corollary, it is also shown the equivalence between a subset of an n-dimensional Euclidean space being compact and being closed and bounded. In the next section, we formalize the definitions of L1-norm (Manhattan Norm) and maximum norm and show their topological equivalence in n-dimensional Euclidean spaces and finite-dimensional real linear spaces. In the last section, we formalize the linear isometries and their topological properties. Namely, it is shown that a linear isometry between real normed spaces preserves properties such as continuity, the convergence of a sequence, openness, closeness, and compactness of subsets. Finally, it is shown that finite-dimensional real normed spaces are proper metric spaces. We referred to [5], [9], and [7] in the formalization.