Finite Dimensional Real Vector Space with Euclidean Norm form Normed Vector Space
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Theorem
Let $\R^n$ be an $n$-dimensional real vector space.
Let $\norm {\, \cdot \,}_2$ be the Euclidean norm.
Then $\struct {\R^n, \norm {\, \cdot \,}_2}$ is a normed vector space.
Proof
We have that:
- By Euclidean Space is Normed Vector Space, $\norm {\, \cdot \,}_2$ is a norm on $\R^n$
By definition, $\struct {\R^n, \norm {\, \cdot \,}_2}$ is a normed vector space.
$\blacksquare$
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.2$: Normed and Banach spaces. Normed Spaces