Normtopologie

Normtopologie, or the norm topology, is the topology on a normed vector space (V, ||·||) generated by the open balls B_r(x) = { y in V : ||y - x|| < r } for r > 0. Equivalently, it is the metric topology induced by the distance d(x,y) = ||x - y||. This topology is translation invariant and Hausdorff, and it is locally convex since the open balls are convex neighborhoods of each point.

In the norm topology, convergence of a sequence x_n to x is characterized by ||x_n - x|| → 0. A function between normed spaces is continuous precisely when it is continuous with respect to this topology; in particular, linear functionals are continuous exactly when they are bounded.

On finite-dimensional spaces, all norms induce the same topology, so the norm topology is essentially unique up to isomorphism in that setting. In contrast, on infinite-dimensional spaces different norms can yield genuinely different topologies.

Completeness is a key aspect: if every Cauchy sequence with respect to the norm converges in V, the space is a Banach space. The norm topology also underpins notions such as open and closed sets, continuity, and convergence for sequences and nets, and it explains many geometric properties such as the behavior of balls and convexity.

Common examples include R^n with the Euclidean norm, l^p spaces with the p-norm, and C([0,1]) with the sup norm. Notions of compactness differ between finite and infinite dimensions: in finite dimensions, closed and bounded sets are compact; in infinite dimensions, bounded sets need not be compact. The norm topology is the standard framework for most of functional analysis.