Finite-dimensional vector-space topology

Every finite-dimensional real vector space has a unique Hausdorff topology making it a topological vector space. Choosing a basis identifies it with ordinary Euclidean space; every linear map between such spaces is continuous.

On a finite-dimensional vector space over a complete valued field, any two norms induce the same topology. Choosing a basis reduces the claim to comparison with the maximum of the absolute values of the coordinates.

For $x\in {\mathbb{R}}^n$, Cauchy-Schwarz inequality applied to $(|x_i|)$ and $(1)$ gives the upper bound. Equal absolute coordinate values attain its factor $\sqrt{n}$. Expanding $({\sum }_i|x_i|{)}^2$ gives the lower bound, and a vector with only one nonzero coordinate attains its factor one. Thus both constants are optimal. These Lp norms define the same finite-dimensional topology despite having different unit balls.