I'm reading through my lecture notes and noticed that I can combine theorems to show det:$ \mathbb{K}^{n \times n} \rightarrow \mathbb{K}$ is continuous (and bounded). Looking at the det function in terms of the sum of entries and corresponding entries of the cofactor matrix it's obvious it is continuous, but I want to know if my thinking is correct so I can apply it to similar situations.
(1) A linear map from a finite dimensional vector space to a normed space is bounded.
(2) Boundedness and continuity are equivalent for a linear map between $2$ normed spaces.
If we regard matrices in $\mathbb{K}^{n \times n}$ as an t-tuple of vectors in $\mathbb{K}^n$, then the determinant function det:$ \mathbb({K}^{n})^{n} \rightarrow \mathbb{K}$ is an n-multilinear function of the row or columns of the matrix. Then viewing this multilinear map as a linear-map-valued linear map, we can apply (1) and (2) to show that it is bounded and continuous.
Not really a question, but like I said I'm just playing around with the theorems and definitions to see if I understand it, so if I have made a mistake please let me know!