Equivalent norms

Two norms on one vector space are equivalent when constants $c,C>0$ satisfyfor every vector. Equivalent norms define the same convergent sequences, open sets, and continuous linear functionals.

Any two norms on a finite-dimensional real or complex vector space are equivalent. Relative to a fixed basis, continuity and positivity of a norm on the compact Euclidean unit sphere give uniform positive lower and finite upper bounds.