Product of Real Matrices

In the setting of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces, let $m,n,p\in\mathbb{N}$, and let $A\in\mathcal{M}_{m\times n}(\mathbb{R})$ and $B\in\mathcal{M}_{n\times p}(\mathbb{R})$ be real matrices, with entries $A_{ij}$ and $B_{jk}$.

(i) The product $AB\in\mathcal{M}_{m\times p}(\mathbb{R})$ is the value at $(A,B)$ of the map of The Matrix Product, Matrix-Vector Product, Transpose, Trace, Identity Matrix and Standard Basis Vectors Exist and Are Unique (i), so that

$$(AB)_{ik}=\sum_{j=1}^{n}A_{ij}B_{jk}\qquad(i\in[m],\ k\in[p]).$$

For matrices, juxtaposition denotes this product, not the pointwise product of Pointwise Operations on Real-Valued Maps (i).