Finite Dimensional Real Vector Space with Euclidean Norm form Normed Vector Space

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:

Real Vector Space is Vector Space

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