# Normed space

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1. The Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ endowed with the Euclidean norm ${\displaystyle \|x\|={\sqrt {\sum _{k=1}^{n}|x_{k}|^{2}}}}$ for all ${\displaystyle x\in \mathbb {R} ^{n}}$. This is the canonical example of a finite dimensional vector space; in fact all finite dimensional real normed spaces of dimension n are isomorphic to this space and, indeed, to one another.
2. The space of the equivalence class of all real valued bounded Lebesgue measurable functions on the interval [0,1] with the norm ${\displaystyle \|f\|=\mathop {{\rm {ess}}\sup } _{x\in [0,1]}|f(x)|}$. This is an example of an infinite dimensional normed space.