The Earlier and the New Versions of the Definitions of Euclidean Space, Real Matrices and Normed Spaces Agree

In the setting of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces.

(i) Euclidean space. For every $n\in\mathbb{N}$, the set $\mathbb{R}^{n}$ of Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (i) is the set $\mathbb{R}^{n}$ of Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (i), and the two versions give the same sum, scalar multiple, negative and difference (Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv), Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv)) and the same origin (Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (v), Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (v)).

(ii) Real matrices. For all $m,n\in\mathbb{N}$, the set $\mathcal{M}_{m\times n}(\mathbb{R})$ of real $m\times n$ matrices of the earlier version is the set of real $m\times n$ matrices of the new version, and the two versions give the same entries (Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (ii), Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (ii)), the same sum, scalar multiple, negative and difference (Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iv), Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iv)) and the same zero matrix (Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (v), Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (v)).

(iii) Normed spaces. For every real vector space $V$ and every function $\lVert\cdot\rVert:V\to\mathbb{R}$, $\lVert\cdot\rVert$ is a norm on $V$ in the earlier version if and only if it is a norm on $V$ in the new version; so $(V,\lVert\cdot\rVert)$ is a real normed space in the earlier version if and only if it is a real normed space in the new one.