Chapter 4 — Norms and distances

Author

Jutho Haegeman

Published

November 3, 2025

Summary

This chapter introduces the concept of a norm, and then discusses at length how this can be used to make sense of limits of sequences of vectors, mostly in infinite-dimensional vector spaces, where there are some surprises and not everything is very intuitive. Then, we discuss the implications for linear maps on such normed vector spaces.

Not covered in class

The final part of section 4.1, on compactness was only briefly discussed (from Theorem 4.10 onwards). Section 4.2 (Banach spaces) was only discussed at high level, and no results need to be known from this section. Subsections 4.4.1 and 4.4.3 were also skipped, but we have covered subsection 4.4.2 on the condition number and how it gives information about the sensitivity or accuracy of solving linear systems.

Important concepts

  • Norm, Hölder p-norm
  • Metric (\(=\) distance), distance in a normed vector space, isometric map, limit of a sequence, continuity of a map, open subset, closed subset, closure, dense subset
  • Equivalence of norms
  • Cauchy sequence, metric completeness (=every Cauchy sequence has limit), Banach space (=metric complete normed vector space), closed subspace, dense subspace, complete set, absolutely converging series, Schauder basis
  • Norm for linear maps, Frobenius norm, bounded map \(\equiv\) continuous map, subordinate norm, submultiplicative norm for operators, induced norm (\(=\) operator norm), spectral radius
  • Condition number

Finite-dimensional vector spaces: all norms are equivalent, always metric complete, finite-dimensional subspaces are always closed

Infinite-dimensional vector spaces: proper infinite-dimensional subspaces of a Banach space can be dense: there is no intuitive or visual way to interpret this concept, there are vectors which are not in this subspace, but nonetheless they are arbitrary close to it (measured using the norm of the surrounding vector space).

Linear maps: continuous if and only if bounded, bounded linear maps with operator norm are metric complete, induced norm is subordinate and submultiplicative, Gelfand formula for spectral radius

For applications / exercises

  • Computing norms

  • Proving inequality relations between standard \(p\) norms or computing induced norms for matrices.

    (But this chapter serves mostly theoretical purposes.)